US20100082643A1 - Computer Implemented Method and Program for Fast Estimation of Matrix Characteristic Values - Google Patents

Computer Implemented Method and Program for Fast Estimation of Matrix Characteristic Values Download PDF

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US20100082643A1
US20100082643A1 US12632062 US63206209A US2010082643A1 US 20100082643 A1 US20100082643 A1 US 20100082643A1 US 12632062 US12632062 US 12632062 US 63206209 A US63206209 A US 63206209A US 2010082643 A1 US2010082643 A1 US 2010082643A1
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matrix
singular
values
term
step
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Jacob Gilmore Martin
Earl Rodney Canfield
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Selective Inc
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Selective Inc
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    • GPHYSICS
    • G06COMPUTING; CALCULATING; COUNTING
    • G06FELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/30Information retrieval; Database structures therefor ; File system structures therefor
    • G06F17/3061Information retrieval; Database structures therefor ; File system structures therefor of unstructured textual data
    • G06F17/30613Indexing

Abstract

A term-by-document (or part-by-collection) matrix can be used to index documents (or collections) for information retrieval applications. Reducing the rank of the indexing matrix can further reduce the complexity of information retrieval. A method for index matrix rank reduction can involve computing a singular value decomposition and then retaining singular values based on the singular values corresponding to singular values of multiple topics. The expected singular values corresponding to a topic can be determined using the roots of a specially formed characteristic polynomial. The coefficients of the special characteristic polynomial can be based on computing the determinants of a Gram matrix of term (or part) probabilities, a method of recursion, or a method of recursion further weighted by the probability of document (or collection) lengths.

Description

    RELATED APPLICATIONS
  • [0001]
    This patent application is a divisional of U.S. patent application Ser. No. 11/505,654 entitled “Selective Latent Semantic Indexing Method for Information Retrieval Applications,” filed Aug. 17, 2006 (now U.S. Pat. No. 7,630,992), which claims priority under 35 U.S.C. §119 to U.S. Provisional Patent Application No. 60/740,700, entitled “Predicting a Topic's Singular Values and Singular Vectors to Provide a New Type of Low Rank Approximation for Information Retrieval Applications,” filed Nov. 30, 2005. The complete disclosure of each of the above identified applications is hereby fully incorporated herein by reference.
  • TECHNICAL FIELD
  • [0002]
    The present invention is generally directed to computer-based information retrieval systems. More particularly, the present invention relates to avoiding the loss of topical coverage and increasing information retrieval performance when using reduced rank models in computer-based information retrieval systems.
  • BACKGROUND OF THE INVENTION
  • [0003]
    Computer-based information systems can store large amounts of data. Despite the potentially enormous size of such data collections, information retrieval queries over a dataset attempt to be as informative, rapid, and accurate as possible. Information retrieval systems often employ indexing techniques to improve precision, improve recall performance and rapidly access specific information within a dataset.
  • [0004]
    Data stored in an information retrieval system for textual data can be indexed using a term-by-document matrix. In that case, “term” means a word or phrase and a “document” is a collection of terms. However, generalized meanings of “term” and “document” can apply, as discussed hereinafter. A term-by-document matrix represents each term as a row and each document as a column. For a column representing a particular document, the elements going down the column can represent some function of the existence of terms within the document. For example, if term A is not used in document B, then the element in a term-by-document matrix that is in both row A and column B could be a zero to represent the absence of the term in the document. Alternatively, if term A is used X times in document B, then the element in a term-by-document matrix that is in both row A and column B could be an X to represent the presence of the term A occurring X number of times in the document.
  • [0005]
    This term-by-document matrix structure enables response to keyword search queries. The row of the term-by-document matrix that corresponds to the queried keyword is examined by the information retrieval system. Elements in that row indicate inclusion of that keyword term within the documents represented by those columns. Such inclusion prompts the information retrieval system to return the documents in response to the keyword query. Thus, the search returns the documents containing a specific keyword by examining a single matrix. Once this term-by-document matrix is constructed, the individual documents within a dataset do not need to be searched when forming a response to a keyword query.
  • [0006]
    Furthermore, the elements of the term-by-document matrix can include a measure of the relevance of the term (given by the row) to the document (given by the column). This measure can be as simple as a count of how many times the term occurs within the document. Likewise, a more involved metric can be employed. Forming a term-by-document matrix with such elements lends to statistical notions for the use of the matrix and enables more detailed query responses. For example, a response to a keyword query can be a list of documents containing a keyword and that list can be ordered such that the documents most relevant to the keyword are listed first. The most relevant documents can be those documents that include the most instances of the keyword.
  • [0007]
    For extremely large sets of documents with multiple keyword terms, the term-by-document matrix can become too large to manipulate during a keyword query. For this reason, simplification techniques can be employed that approximate the term-by-document matrix with a simpler matrix that is less time consuming to manipulate. Conventional Latent Semantic Indexing (LSI) employs a reduced rank version of the term-by-document matrix as an approximation of the original matrix. The approximation obtained has also been shown to be useful in increasing the overall information retrieval performance.
  • [0008]
    The LSI approach seeks to factor the term-by-document matrix using Singular Value Decomposition (SVD) and then makes some of the smallest singular values equal to zero, thereby leaving a reduced rank approximation of the term-by-document matrix. To achieve an approximation of the term-by-document matrix that is of reduced rank k, the conventional LSI approach only retains the k largest singular values and sets all of the other singular values to zero. The resultant matrix is an approximation of the original term-by-document matrix but with a lower rank of k (i.e., including only the k largest singular values).
  • [0009]
    Generating a reduced rank approximation of the term-by-document matrix is useful for reducing the computational complexity of indexed information retrieval. It is also said that it can produce a matrix that can be considered less “noisy.” Such a reduced rank matrix also can retrieve related term entries that would have been excluded based on the original term-by-document matrix due to synonymy. The reduced rank matrix can associate words that never actually appear together in the same document.
  • [0010]
    Such rank reduction is not lossless. Making some of the singular values equal to zero reduces the rank of a matrix and invariably removes some information. When using the conventional LSI techniques, one example of a loss that can be introduced is a loss of topical coverage. A topic is generally conceptualized as a subject addressed within the documents of the dataset. Mathematically, a topic can be considered a probability distribution over all terms. For example, the term “hexagon” is perhaps more probabilistically likely to be related to a topic of a mathematical nature than it would be to a topic of a historical nature.
  • [0011]
    Conventional LSI rank reduction does not always maintain coverage of all topics. The blind selection of the k largest singular values can result in the removal of information that loses the connection between a topic and certain keywords. Retaining only the largest singular values can allow the term-document relationships of more common topics to dominate the reduced rank matrix at the cost of the removal of the term-document relationships of less frequently represented topics.
  • [0012]
    Thus, there is a need in the art for a rank reduction technique that retains the general benefits of the conventional LSI approach while attempting to maintain topical coverage during rank reduction of the term-by-document indexing matrix. More particularly, a need exists in the art for selectively identifying the singular values of interest related to a dataset that has been annotated and stored in some matrix format.
  • SUMMARY OF THE INVENTION
  • [0013]
    The present invention discloses a computer system and program product including an algorithm for the fast computation of a characteristic value of a matrix. The computer system includes an algorithm for computing a characteristic value of a matrix comprising the steps of providing a first matrix, extrapolating a probabilistic matrix model from the first matrix, and using the probabilistic matrix model to obtain a characteristic value of the first matrix.
  • [0014]
    The inventive method of Selective Latent Semantic Indexing (SLSI) comprises a technique for reducing the rank of a term-by-document matrix in a way that can reduce or prevent the loss of topical coverage and that can give control over the amounts of each topic to cover. The method can begin by forming a term-by-document matrix and expanding the matrix via Singular Value Decomposition (SVD). However, instead of simply retaining the largest singular values, the SLSI method can determine which singular values to retain using information about how the singular values relate to the topics. Identifying the singular values that correspond to particular topics can enable the selection of specific singular values to retain and to abandon so as to retain any possible coverage of all topics of interest spanned by the documents.
  • [0015]
    The SLSI method can establish a partition of the singular values into groups corresponding to each topic. This partitioning can be performed using a technique for estimating the singular values that correspond to each topic. The estimated singular values approximate the singular values obtained from the SVD to identify a mapping between the SVD singular values and the topics. Once the singular values are partitioned by topic, the selection of singular values to retain in the reduced rank term-by-document matrix can be made with knowledge of the partitioning.
  • [0016]
    One skilled in the art will appreciate that references to terms and documents are non-limiting examples. The inventive method for indexing terms and documents can just as well address indexing of genes within individuals; atoms within molecules; elements within sets; or any general parts within collections. One skilled in the art will appreciate that such generalization from terms within documents to the indexing of parts within collections occurs without departing from the scope or spirit of the invention.
  • [0017]
    The method of Selective Latent Semantic Indexing presented in this summary is for illustrative purposes only. Various aspects of the present invention can be more clearly understood and appreciated from a review of the following detailed description of the disclosed embodiments and by reference to the drawings and any claims that follow. Moreover, other aspects, systems, methods, features, advantages, and objects of the present invention will become apparent to one with skill in the art upon examination of the following drawings and detailed description. It is intended that all such aspects, systems, methods, features, advantages, and objects are included within this description, are within the scope of the present invention, and are protected by any accompanying claims.
  • BRIEF DESCRIPTION OF THE DRAWINGS
  • [0018]
    FIG. 1 is a logical flow diagram depicting a method for selectively generating a reduced rank matrix approximation for information retrieval according to an exemplary embodiment of the invention.
  • [0019]
    FIG. 2 is a logical flow diagram depicting a method for selecting k singular values based on their correspondence to singular values of multiple topics according to an exemplary embodiment of the invention.
  • [0020]
    FIG. 3 is a logical flow diagram depicting a method for estimating the singular values of a topic according to an exemplary embodiment of the invention.
  • [0021]
    FIG. 4 is a logical flow diagram depicting a method for generating characteristic coefficients to estimate singular values corresponding to a topic according to an exemplary embodiment of the invention.
  • [0022]
    FIG. 5 is a logical flow diagram depicting a method for recursively generating characteristic coefficients to estimate the singular values corresponding to a topic where the document lengths are uniform according to an exemplary embodiment of the invention.
  • [0023]
    FIG. 6 is a logical flow diagram depicting a method for recursively generating characteristic coefficients to estimate the singular values corresponding to a topic where the document lengths are non-uniform according to an exemplary embodiment of the invention.
  • [0024]
    FIG. 7 is a functional block diagram depicting an Internet search engine system according to an exemplary embodiment of the invention.
  • DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS
  • [0025]
    The inventive information retrieval method can comprise steps for selecting certain singular values to retain in a reduced rank approximation of a term-by-document indexing matrix. The selection of singular values can comprise steps for ensuring that all desired topics covered by the documents are retained after the rank reduction process. The inventive rank reduction method can ensure this retention of topical coverage by using information identifying the singular values that correspond to certain topics. For example, when selecting which singular values to retain, this information can be used to ensure that at least one singular value is retained corresponding to each desired topic.
  • [0026]
    The information identifying the singular values corresponding to certain topics can be obtained by estimating the singular values of each topic and then identifying the actual singular values of the term-by-document matrix that are closest to the estimated singular values. This process can be repeated for all desired topics. Once the matching is performed between the estimated singular values for each topic and all of the actual singular values of the term-by-document matrix, the result is a partitioning, by topic, of the actual term-by-document singular values. Then, the selection of singular values to retain in the reduced rank term-by-document matrix can be made. This partitioning method can allow retaining any combination of singular values from each desired partition, the partition being desired if loss of coverage of the topic corresponding to that partition is undesirable.
  • [0027]
    A number of methods can be employed to select the singular vectors that correspond to a particular estimated singular value. These include choosing multiple singular vectors for one expected singular value. For instance, if an estimated singular value is selected, there could be several actual singular vectors that correspond to actual singular values that are close to the estimated singular value. In this case, one, some, or even all of the candidate actual singular vectors can be selected.
  • [0028]
    Estimating the singular values of a topic can be carried out by finding the roots of a specially formed characteristic equation. The coefficients of the characteristic equation can be calculated explicitly. Alternatively, the coefficients can be calculated more efficiently using a recursion method. It is also possible to calculate the coefficients of the characteristic equation in a manner that probabilistically weights the contribution from documents of varying length.
  • [0029]
    The inventive singular value estimation method can estimate the singular values of term-by-document matrices that are based on various term metrics. One example of such a metric uses “0” and “1” by placing a “1” in the matrix if the term exists at least once in the corresponding document and placing a “0” in the matrix if the term is not present in the document. Another example of such a metric uses “−1” and “1” by placing a “1” in the matrix if the term exists at least once in the corresponding document and placing a “−1” in the matrix if the term is not present in the document. Yet another example metric uses a term frequency technique where the value of each element in the matrix represents how often the corresponding term appears within the corresponding document. This value can represent an absolute count of the occurrences of the term, it can represent the logarithm of an absolute count of the occurrences of the term, it can represent the proportionality of the occurrence among all of the terms, or it can represent a weighted proportion. Various other term-by-document matrix formations will be apparent to those of ordinary skill in the art. The inventive method can operate with any representation of the term-by-document matrix without departing from the scope or spirit of the invention.
  • [0030]
    One of ordinary skill in the art will appreciate that embodiments of the inventive method can comprise software or firmware code executing on a microcontroller, microprocessor, or DSP processor; state machines implemented in application specific or programmable logic; or numerous other forms without departing from the spirit and scope of the invention. The invention can be provided as a computer program which can include a machine-readable medium having stored thereon instructions which can be used to program a computer (or other electronic devices) to perform a process according to the invention.
  • [0031]
    The machine-readable medium can include, but is not limited to, floppy diskettes, optical disks, CD-ROMs, and magneto-optical disks, ROMs, RAMs, EPROMs, EEPROMs, magnetic or optical cards, flash memory, or other type of media or machine-readable medium suitable for storing electronic instructions.
  • [0032]
    Certain steps in the processes or process flows described in all of the logic flow diagrams referred to below must naturally precede others for the invention to function as described. However, the invention is not limited to the order of the steps described if such order or sequence does not alter the functionality of the present invention. That is, it is recognized that some steps can be performed before, after, or in parallel with other steps without departing from the scope and spirit of the present invention.
  • [0033]
    Further, one of ordinary skill in the art would be able to write such a computer program or to identify the appropriate hardware circuits to implement the disclosed invention without difficulty based on the flow charts and associated description in the application text, for example. Therefore, disclosure of a particular set of program code instructions or detailed hardware devices is not considered necessary for an adequate understanding of how to make and use the invention. The inventive functionality of the claimed computer implemented processes will be explained in more detail in the following description in conjunction with the remaining figures illustrating other logical process flows.
  • [0034]
    Turning now to the drawings, in which like reference numerals refer to like elements or steps, FIG. 1 is a logical flow diagram depicting a method 100 for selectively generating a reduced rank matrix approximation for information retrieval according to an exemplary embodiment of the invention. In initial step 105, a corpus model is identified. The corpus model is a probabilistic matrix model representing the probability of all terms within all topics, the probability of a topic appearing in a document collection, the document length probabilities within the topics, and the type of function used to create the term-by-document matrix. This corpus model can be the source of the probability distributions used in this method.
  • [0035]
    The corpus model can be identified by estimating the model or the method can be provided the model as an input. Given a document collection, realistic simplifications can be made about the corpus model underlying it. For example, topics will usually not have the same number of document representatives. Some documents in a given collection will be more or less popular. In addition, any given topic will usually have some terms that are much more common within that topic than other terms are.
  • [0036]
    Random sampling and trivial data analysis techniques, apparent to one skilled in the art, lead to the ability to make realistic assumptions about the approximate parameters of a corpus model that would generate a given collection of documents. For example, an algorithm can discover which terms correspond to disjoint topics by using techniques familiar to those skilled in the art of information retrieval. One example method for discovering the topics can be to compute several reduced rank-k LSI approximations of a term-by-document matrix A to find the sets of terms that are used most similarly across a collection of documents. Other techniques for determining the terms that are in each topic will be apparent to one of ordinary skill in the art. Term frequency counts within a particular topic can be calculated which will give an estimate of the term distributions within a topic. Document lengths and topic probabilities can also be estimated by sampling, giving every parameter of a corpus model that could have generated a collection of documents. With this information and the inventive SLSI process, the expected singular values that correspond to each topic can be identified with high probability.
  • [0037]
    In step 110, a term-by-document matrix A is formed. This matrix is an indexing matrix that can be used to identify the documents containing a certain term or terms. The matrix A can be formed by explicitly examining each term in each document of the set of documents to be searched. Alternatively, an exemplary matrix A can be formed from a statistical model of the type of documents contained in the document set. The inventive method 100 can be used to produce and employ a reduced rank approximation of this matrix for information retrieval.
  • [0038]
    In step 120, the singular values of the matrix are computed. These singular values can be computed using singular value decomposition (SVD), which yields the singular values together with their corresponding singular vectors. When the outer product of the singular vectors of a matrix are each multiplied by their corresponding singular value and then summed together, the result is the original matrix. In other words, the result of the SVD is the decomposition of a matrix into its component singular vectors such that the sum of the rank one matrices given by the outer products of the singular vectors multiplied by their corresponding singular values is the original matrix.
  • [0039]
    In step 130, a quantity k of singular values is selected to retain in a reduced rank matrix A k based on the singular values corresponding to the computed expected singular values of multiple topics. The selected k singular values will be used to form an approximation of the matrix A having a reduced rank of k, as discussed in more detail hereinafter with reference to step 150. The singular values can be selected based on the expected singular values of one or more topics. Exemplary steps for subroutine 130 are described hereinafter with reference to FIG. 2. The reduced rank matrix is referred to in the figures as A k .
  • [0040]
    In step 140, the non-selected singular values are set to zero. Setting the non-selected singular values to zero effectively cancels the corresponding singular vectors related to those values from the term-by-document matrix A. With those values canceled, the reduced rank matrix A k can be calculated.
  • [0041]
    Then, in step 150, the reduced rank matrix A k is computed based on the k selected singular values. In this step, the reduced rank matrix A k is generated by recombining the k selected singular values and their corresponding singular vectors to compute the matrix A k , which is a reduced rank approximation of the matrix A.
  • [0042]
    Since A k may be prohibitively large, it is also possible to use only the k selected singular vectors to perform data retrieval. In such an implementation, the reduced rank matrix may not be completely recombined, but instead the information from the matrix is used in its decomposed form.
  • [0043]
    In step 160, information retrieval is performed using the matrix A k in response to a query. Thus, queries to retrieve information from the original dataset are processed using the reduced rank approximation matrix A k or the k selected singular vectors instead of the original term-by-document matrix A.
  • [0044]
    One of ordinary skill in the art will appreciate that certain steps of this method can be reusable across repeated applications or iterations of the method 100 without departing from the scope or spirit of the present invention. For example, the reduced rank matrix A k developed in step 150 can be used repeatedly to process multiple information retrieval queries in step 160. Also, the singular values that are established for each of the topics in step 130 can be retained and applied to a new formation of matrix A to partition, by topic, the singular values of the new matrix A. In other words, a new matrix A formed in a later iteration of step 110 and its singular values computed in a later iteration of step 120 can use, in step 130, the same topical sets of singular values that were established in an earlier iteration of step 130 of the method 100. Essentially, the methods that calculate the estimated singular values in step 220 can be used for future matrices drawn from the same corpus model as in an earlier invocation of these steps.
  • [0045]
    FIG. 2 is a logical flow diagram depicting a method 130 for selecting a quantity k of singular values to retain in a reduced rank matrix A k based on the singular values corresponding to the computed singular values of multiple topics according to an exemplary embodiment. This method 130 is an exemplary embodiment of the subroutine 130 within the method 100 described with reference to FIG. 1. This exemplary method 130 can establish the correspondence between the singular values of the term-by-document matrix A and the topics covered by those documents. This correspondence can be accomplished by estimating the singular values of the topics. Other exemplary embodiments of subroutine 130, not using topical prediction, can be employed to select the singular values to retain based on their topical relationships. For example, the singular values corresponding to one or more topics may be known from models of the data set, other estimation procedures, or empirical calculations made directly on the actual data set or model data sets.
  • [0046]
    In step 210, a topic can be selected from the set of desired topics. The desired topics can comprise all topics covered by the documents or the desired topics can comprise a subset of topics such that only those topics desired within the reduced rank approximation of the term-by-document matrix are used for selecting the k singular values of the matrix A. Step 210 and decision step 250 form a loop over these desired topics. For each iteration of this loop, steps 220, 230 and 240 can be performed for one of the topics.
  • [0047]
    In step 220, the singular values of the topic are estimated. This prediction is represented as a subroutine, exemplary steps of which are described hereinafter with reference to FIG. 3.
  • [0048]
    In step 230, the estimated singular values of the topic are compared to the actual singular values of the term-by-document matrix A to determine the actual singular values that correspond to the topic. The method of comparison can comprise finding the actual singular values of matrix A that are closest to the singular values estimated for the topic.
  • [0049]
    In step 240, the identification of actual singular values of matrix A that correspond to the estimated singular values for the topic are used to establish a topical partitioning of the singular values of matrix A. In other words, actual singular values that correspond to the estimated singular values for the topic can be identified and grouped to partition the selected topic from other topics represented in the matrix A. As the steps 220-240 are performed for multiple topics, the singular values of the term-by-document matrix A are partitioned into subsets of singular values that correspond to the topics covered by the documents.
  • [0050]
    In step 250, the method 130 determines whether the partitioning is complete for all desired topics. If not, then the method 130 branches back to step 210 to partition another topic. If yes, then the method 130 branches to step 260.
  • [0051]
    In step 260, the number of singular values to retain from each topical partition of the singular values of A is determined. The total number of these retained singular values is k (or less), where k is the desired rank of the reduced rank matrix A k . In an exemplary embodiment, determining the number of singular values to retain from each topical partition can comprise taking the largest singular values from each topic partition in a quantity such that each topic is represented by a substantially equal number of singular values. Alternatively, determining the number of singular values to retain from each topical partition can comprise selecting different numbers of singular values from each topic. For example, this selection can be done by weighting the number of singular values selected from any given topic. This weighting can be made by the relevance of the topic, or the size of the topic, or by some other metric of topical importance. One of ordinary skill in the art will appreciate the application of various approaches to allocating the k singular values between the topics to establish representative distributions of singular values over the desired topics. Such alternative embodiments are within the scope and spirit of the present invention.
  • [0052]
    In step 270, the k singular values of A that are to be retained in the reduced rank approximation are selected. The numbers of singular values to retain from each topical partition that were determined in step 260 guide the selection of the particular k singular values. For example, if the method 130 determined in step 260 to retain the largest three singular values for each topic, then those singular values are selected for retention in step 270.
  • [0053]
    From step 270, the method 130 proceeds to step 140 (FIG. 1).
  • [0054]
    FIG. 3 is a flow chart depicting a method 220 for calculating the expected singular values corresponding to a topic according to an exemplary embodiment, as referred to in step 220 of FIG. 2. In step 310, a quantity t of partial characteristic coefficients c1 through ct are generated, where t is the number of terms in the current topic. These t values are used in the calculation of partial coefficients of the terms in a specially formed characteristic polynomial detailed below in Equation 1. These cn values only represent partial coefficients because they are divided by a factorial to form the coefficients. A characteristic polynomial can be said to represent a matrix, or parameters thereof, in a polynomial form. Exemplary embodiments of step 310 will be described in more detail hereinafter with reference to FIGS. 4-6.
  • [0055]
    In step 320, the partial characteristic coefficients generated in step 310 are used to construct a specially formed characteristic polynomial illustrated in Equation 1 below.
  • [0000]
    λ t - c 1 1 ! λ t - 1 + c 2 2 ! λ t - 2 - ± c t t ! λ 0 ( 1 )
  • [0056]
    In step 330, the roots of the polynomial are computed. One of ordinary skill in the art will appreciate various methods for computing the roots of a polynomial.
  • [0057]
    In step 340, one of the roots computed in step 330 is selected.
  • [0058]
    In step 350, the selected root is multiplied by the number of documents within the topic. If the information on the documents is from a statistical model rather than actual documents, this number of documents can be the expected number of documents within the topic. That is, if the documents within the data set are known and classified by topic then the number of documents within a topic can be counted, otherwise the expected number of documents within a given topic can be extracted by sampling or derived from a probabilistic model of the data.
  • [0059]
    In step 360, the square-root of the result obtained from step 350 is computed. The result from step 360 is one of the estimated singular values for the given topic.
  • [0060]
    Next, decision step 370 tests if any additional roots remain to be operated upon. If so, the method 220 branches back to step 340 where another root is selected to be operated on by steps 350 and 360. If not, then all of the roots have been operated on. These operations result in the full set of estimated singular values for the topic. Once this completion is determined, decision step 370 transitions to step 230 (FIG. 2).
  • [0061]
    Using method 220, the estimated singular values can be obtained for term-by-document matrices based upon any one of (0,1) elements, (−1,1) elements, various types of term frequency or term count representations, or any combinations, scaling, or functional transformations thereof. Thus, method 220 can be generalized for use with different forms of the term-by-document matrix.
  • [0062]
    FIG. 4 is a flow chart depicting a method 310A for generating partial characteristic coefficients to estimate the singular values corresponding to a topic as referred to in step 310 of FIG. 3, according to an exemplary embodiment.
  • [0063]
    In step 410, a coefficient index i is selected from the range of one to t (inclusive of both one and t), where t is the number of terms in the current topic.
  • [0064]
    In step 420, a vector v is formed with t elements representing the probabilities of the terms in the topic. For example, if a first term is more likely to occur within a topic than a second term, then the element of vector v corresponding to the first term will be larger than the element of vector v corresponding to the second term.
  • [0065]
    In step 430, i copies of vector v are arranged as the columns of a matrix B that has quantity t rows and quantity i columns.
  • [0066]
    In step 440, the matrix transpose of B is multiplied by matrix B to form a Gram matrix, written as BTB. The transpose of a first matrix is a second matrix, produced by turning the rows of the first matrix into the columns of the second matrix as to reflect the matrix about the diagonal that runs from the top left to bottom right of the matrix. In step 450, the determinant of the Gram matrix BTB is computed to identify the ith characteristic coefficient. In other words, for the index i selected in step 410, the determinant computed in step 450 is the ith characteristic coefficient, written as ci.
  • [0067]
    Next, decision step 460 determines whether any additional indexes remain to be operated upon. If so, the loop beginning at step 410 is repeated for the next remaining index i. This loop is repeated until all indexes (one through t inclusive) have been operated upon. The results of the indexed iterations are the t partial characteristic coefficients used by method 220 in FIG. 3. The method 310A then proceeds to step 320 (FIG. 3).
  • [0068]
    FIG. 5 is a flow chart depicting a method 310B for recursively generating partial characteristic coefficients to estimate the singular values corresponding to a topic where the document lengths are uniform, as referred to in step 310 of FIG. 3, according to an alternative exemplary embodiment. Recursion refers to generating partial characteristic coefficients such that each coefficient is based on those coefficients already computed.
  • [0069]
    While the resultant coefficients of method 310B may be equivalent to those generated by the exemplary method 310A described with reference to FIG. 3, the recursive approach can provide more accurate estimations and can be more computationally efficient.
  • [0070]
    In step 510, a square matrix M with t rows and t columns is formed by inserting the expected value of the product of the ith and jth terms into the ith and jth column of M. For example, in Equation 2, each element of the matrix M is formed such that i is the row of that element and j is the column of that element. In Equation 2, l corresponds to the length of the documents and pi corresponds to the probability of the ith term in the current topic being processed.
  • [0071]
    Equation 2 corresponds to an exemplary covariance expectation expression for the case where the term-by-document matrix is generated using term frequency counts. It should be noted that for any given matrix generation technique, only the matrix M need be changed. Therefore, by altering the M matrix in step 510, the estimated singular values can be obtained for term-by-document matrices based upon any one of (0,1) elements, (−1,1) elements, various types of term frequency or term count representations, or any combinations, scaling, or functional transformations thereof. Thus, method 220 can be generalized for use with different forms of the term-by-document matrix. Several alternative example formulations of the M matrix are addressed hereinafter.
  • [0000]
    M ij = { ( l ( l - 1 ) p i 2 + lp i ) , i = j l ( l - 1 ) p i p j , i j ( 2 )
  • [0072]
    In step 520, the sequence an is computed according to the formula illustrated in Equation 3 below, where n ranges from one to t (inclusive of both one and t).
  • [0000]

    a n=trace(M n)  (3)
  • [0073]
    Each value in the sequence, an, is computed by raising the matrix M to the nth power and taking the trace of the resulting matrix as shown in Equation 3. One skilled in the art will recognize efficient techniques for accomplishing this task. One such technique is to sum the nth powers of the eigenvalues of M. The trace of a square matrix is defined to be the sum of the elements on the main diagonal of the matrix. The main diagonal of a square matrix is the diagonal from the upper left to the lower right.
  • [0074]
    In step 530, a coefficient of index zero, written as c0, is initialized to the value of one. This coefficient forms the base of the recursion performed in step 540. Then, in step 540, the remaining coefficients up to ct are computed using the recursion illustrated in Equation 4 below.
  • [0000]
    c n + 1 = j = 0 n ( n j ) ( - 1 ) j ( j ! ) a j + 1 c n - j ( 4 )
  • [0075]
    The recursion Equation 4 can compute the value of the coefficient cn+1 using the value of the nth and lower coefficients. Thus, c1 is computed using c0, c2 is computed using c0 and c1, and so on up to ct. The results of this recursion are the t partial characteristic coefficients used by method 220 in FIG. 3. From step 540, the method 310B proceeds to step 320 (FIG. 3).
  • [0076]
    FIG. 6 is a flow chart depicting a method 310C for recursively generating partial characteristic coefficients to compute the expected singular values corresponding to a topic where the document lengths are not uniform, as referred to in step 310 of FIG. 3, according to another alternative exemplary embodiment. The method 310C can be considered a generalization of method 310B. While the method of 310B computes the partial characteristic coefficients when all of the documents are the same length, method 310C can compute the coefficients when the document lengths comprise a finite probability distribution. The lengths are the number of terms in a document and the probability distributions on lengths for each topic can come from the Corpus Model or can be estimated by sampling.
  • [0077]
    In step 610, a possible document length is selected.
  • [0078]
    In step 620, a square matrix M with t rows and t columns is formed according to the type of term-by-document matrix that is being analyzed. For example, a formula shown in Equation 2 above gives the formula for the entries when the term-by-document matrices are formed using term frequencies. As mentioned previously, any other type of matrix formulation technique could be substituted for this step by simply changing the way M is constructed. Several alternative example formulations of the M matrix are addressed hereinafter.
  • [0079]
    In step 630, the sequence an, is computed where n ranges from one to t (inclusive of both one and t). Each value in the sequence an can be computed by raising the matrix M to the nth power and taking the trace of the resulting matrix as shown in Equation 3. One skilled in the art will recognize efficient techniques for accomplishing this task. One exemplary technique is to sum the nth powers of the eigenvalues of M.
  • [0080]
    In step 640, a coefficient of index zero, written as c0(l), is initialized to the value of one. This coefficient forms the base of the recursion performed in step 650. Then in step 650, the remaining coefficients up to ct(l) are computed using the recursion illustrated in Equation 5 below.
  • [0000]
    c n + 1 ( l ) = j = 0 n ( n j ) ( - 1 ) j ( j ! ) a j + 1 c n - j ( l ) ( 5 )
  • [0081]
    The recursion Equation 5 can compute the value of the coefficient cn+1(l) using the value of the nth and lower coefficients. Thus, c1(l) is computed using c0(l), c2(l) is computed using c0(l) and c1(l), and so on up to ct(l). The results of this recursion are the t partial characteristic coefficients for the possible length given by l.
  • [0082]
    From step 650, decision step 660 determines if all possible lengths have been used for calculating a set of coefficients. If possible lengths remain, the method 310C loops back to step 610 where another possible length is selected for use in steps 620, 630, 640 and 650. If there are no possible lengths remaining to iterate, then the method 310C continues to step 670.
  • [0083]
    In step 670, the partial characteristic coefficients for each length (generated at each pass of step 650) can be combined together as a weighted sum according to the expression set forth in Equation 6 below.
  • [0000]
    c i = l c i ( l ) prob ( l ) ( 6 )
  • [0084]
    The weight of each term in the sum is the probability of a document of length l appearing within the topic. These probabilities can be derived by sampling, or can be given as inputs from the Corpus Model. The result of this weighted sum is one of the t partial characteristic coefficients used by method 220 in FIG. 3. From step 670, the method 310C proceeds to step 320 (FIG. 3).
  • [0085]
    As mentioned previously, the technique for forming the covariance expectation matrix M is based on the format of the term-by-document matrix A. For example, the formula shown in Equation 2 previously provides the formula for the entries of matrix M when the term-by-document matrix A is formed using term frequencies. Equation 7 below provides a format for elements of the M matrix when the term-by-document matrix elements are either zero or one indicating the presence or absence of a term in a document and when all terms are equally likely. In Equation 7, t is the number of terms, l is the length of the documents, and S(n,k) are the Stirling numbers of the second kind. Stirling numbers are sets of numbers well established in the field of combinatoric mathematics and known to one of ordinary skill on the art.
  • [0000]
    M ij = { l - t - l ( t - 1 ) l , i = j t l - 2 ( t - 1 ) l + ( t - 2 ) l t l = k = 0 t - 3 ( t - 3 k ) ( t - k - 1 ) ! S ( l + 1 , t - k ) t l , i j ( 7 )
  • [0086]
    Equation 8 below provides a format for elements of the M matrix when the term-by-document matrix elements are either zero or one indicating the presence or absence of a term in a document and when the terms each have their own corresponding probabilities. Here pi corresponds to the probability of the ith term in the current topic.
  • [0000]
    M ij = { 1 - ( 1 - p i ) l , i = j 1 - ( 1 - p i ) l - ( 1 - p j ) l - ( 1 - p i - p j ) l , i j ( 8 )
  • [0087]
    Equation 9 below provides a format for elements of the M matrix when the term-by-document matrix elements are either +1 or −1 indicating the presence or absence of a term in a document respectively.
  • [0000]
    M ij = { 1 , i = j 1 - 2 ( 1 - p i ) l - 2 ( 1 - p j ) l + 4 ( 1 - p i - p j ) l , i j ( 9 )
  • [0088]
    Equation 10 below provides a format for elements of the M matrix when the term-by-document matrix elements are the logarithm of the frequency of the term in a document. In Equation 10, ji corresponds to the number of times the ith term is chosen.
  • [0000]
    M ij = [ j 1 + + j t = l ] log j i log j j ( l j 1 j t ) p 1 j 1 p t j t ( 10 )
  • [0089]
    Similarly, Equation 11 below provides a general, non-simplified, format for elements of the M matrix. In Equation 11, ji corresponds to the number of times the ith term is chosen.
  • [0000]
    M ij = [ j 1 + + j t = l ] j i j j ( l j 1 j t ) p 1 j 1 p t j t ( 11 )
  • [0090]
    FIG. 7 is a functional block diagram depicting an Internet search engine system according to an exemplary embodiment of the invention. An exemplary embodiment of the present invention can include one or more computer programs or computer-implemented methods that implement functions or steps described herein and illustrated in the exemplary flowcharts above. The computer program embodiment depicted in FIG. 7 is exemplary and is not intended to limit the inventive indexing method which can also be applied to genetics, bioinformatics, financial data, social science data, or numerous other data mining or information retrieval applications. Such applications can reside and operate on stand-alone, multi-processor, parallelized-processor, distributed, or networked system.
  • [0091]
    A search engine server 710 is connected to the Internet 720 over one or more communication links 715. The communication link 715 can include a path through an intranet, a private network, a telephony network, an Internet protocol (“IP”) network, a packet-switched network, a circuit-switched network, a local area network (“LAN”), a wide area network (“WAN”), a metropolitan area network (“MAN”), the public switched telephone network (“PSTN”), a wireless network, a cellular system, an Ethernet network, or an ATM network, for example. The communication link 715 also can include a signal path that is optical, fiber optic, wired, wireless, wire-line, waveguided, satellite-based, synchronous, asynchronous, or isochronous to name a few possibilities, for example. Signals transmitting over the link 715 can carry or convey data or information digitally or via analog transmission. Such signals can include modulated electrical, optical, microwave, radiofrequency, ultrasonic, or electromagnetic energy, among other energy forms.
  • [0092]
    Search engine server 710 can scan information made available on other servers on the Internet (these other servers are not shown in FIG. 7). Search engine server 710 can form one or more term-by-document matrices A. Such a matrix A can represent the occurrence of terms in documents stored on the other servers attached to Internet 720. By examining the matrix A, search engine server 710 can respond to search queries. Such queries may request which documents on other servers contain a certain term. Such a keyword search query is typical of Internet search engine requests. These keyword searches can be issued to the search engine server 710 through the Internet 720. The searches can originate from user systems 760 that are connected to the Internet 720 by communication links 750.
  • [0093]
    Search engine server 710 may implement the method 100 detailed above. Use of this method can allow the search engine server 710 to reduce the complexity of its indexing matrix A so that keyword search queries may be serviced more efficiently. Method 100 can be used by search engine server 710 to reduce the rank of the indexing matrix without loss of the ability to respond to keyword search queries on particular topics. One of ordinary skill in the art will appreciate how the foregoing reduction in rank of an indexing matrix can be applied to an Internet search engine application as well as numerous other applications of data mining or information retrieval.
  • [0094]
    One of ordinary skill in the art also will appreciate that any of the foregoing techniques can provide indexing on terms or on documents. While the topic based estimations discussed above find clusters of documents, the same techniques can be used to find clusters of terms. Furthermore, some exemplary implementations of such methods may use both term and document clustering. These uses would not depart from the spirit or scope of the present invention.
  • [0095]
    Normalization steps that are common in matrix processing techniques are typically performed before and/or after matrix rank reductions. One of ordinary skill in the art will appreciate that any such normalization, scaling, filtering. thresholding, or reformatting of the results or intermediate values obtained or used by the inventive methods or systems do not depart from the spirit or scope of the present invention.
  • [0096]
    Throughout all foregoing details of the inventive method, reference is made to terms, documents, term-by-document matrixes, topics, and corpus models. Use of this language is merely exemplary and is not intended to limit the information retrieval applications of the inventive methods to the realm of documents made up of words or terms. One of ordinary skill in the art will appreciate, without departure from the scope or spirit of the invention, the generalization of these terms to other information retrieval applications. For example the usage (term, document, topic, corpus model) can be generalized respectively to (element, collection, probabilistic distribution on elements, probabilistic matrix model). This generalization can be re-applied in various application realms, for example, in genetics or bioinformatics the usage can be (gene, individual, expression, population model). An example from the realm of chemistry can be (atom, molecule, classification of chemical, molecular model). Any number of other examples can be drawn from the fields of science, social-sciences, finance, economics, computer science, or otherwise where information representing collections of elements can be indexed and sets of collections categorized by an analogy to the notion of topics. These varied applications of the inventive method are within the scope and spirit of the invention as the invention is not to be limited by any use herein of exemplary language from the realm of term and document indexing or otherwise.
  • [0097]
    Although specific embodiments of the present invention have been described herein in detail, the description is merely for purposes of illustration. The exemplary methods described herein are merely illustrative and, in alternative embodiments of the invention, certain steps can be performed in a different order, performed in parallel with one another, or omitted entirely, and/or certain additional steps can be performed without departing from the scope and spirit of the invention. Additionally, various modifications of, and equivalent steps corresponding to, the disclosed aspects of the exemplary embodiments, in addition to those described herein, can be made by those skilled in the art without departing from the spirit and scope of the present invention defined in the following claims, the scope of which is to be accorded the broadest interpretation so as to encompass such modifications and equivalent structures.

Claims (41)

  1. 1-31. (canceled)
  2. 32. A program product including a computer readable computer program encoded in a storage medium, the computer program executing an algorithm for computing a characteristic value of a matrix, comprising steps of:
    providing a first matrix;
    providing a probability model by sampling from the first matrix; and
    extrapolating from the probability model a characteristic value of the first matrix.
  3. 33. The program product according to claim 32, wherein the characteristic value is a singular value of the first matrix.
  4. 34. The program product according to claim 32, wherein the characteristic value is an eigenvalue of the first matrix.
  5. 35. The program product according to claim 32, wherein the probability model comprises:
    a probability distribution corresponding to a set of elements;
    a probability distribution corresponding to a set of sample lengths; and
    a probability of a sample from the probability model.
  6. 36. The program product according to claim 32, wherein the step of extrapolating from the probability model comprises the steps of:
    constructing a polynomial corresponding to the probability model; and
    extrapolating from the polynomial to obtain the characteristic value of the first matrix.
  7. 37. The program product according to claim 36, wherein the step of extrapolating from the polynomial comprises the steps of:
    finding a root of the polynomial for the probability model of the first matrix; and
    extrapolating from the root of the polynomial to obtain the characteristic value of the first matrix.
  8. 38. The program product according to claim 37, wherein the step of extrapolating from the root of the polynomial comprises the steps of:
    multiplying the root by an expected number of samples from the probability model; and
    taking the square root of the resulting value to obtain the characteristic value.
  9. 39. The program product according to claim 36, wherein the step of constructing the polynomial comprises the steps of:
    computing one or more coefficients ci; and
    dividing each computed coefficient ci by i! to obtain the polynomial
    λ t - c 1 1 ! λ t - 1 + c 2 2 ! λ t - 2 - ± c t t ! λ 0
    wherein each of the non-computed coefficients is set to zero and t is from the probability model.
  10. 40. The program product according to claim 39, wherein the step of computing a coefficient ci of the polynomial comprises computing a probabilistically weighted value
    c i = l c i ( l ) prob ( l )
    comprising an additional coefficient ci(l) corresponding to a length l and probability prob(l) from the probability model to obtain the coefficient ci.
  11. 41. The program product according to claim 40, wherein the step of computing the coefficient ci(l) of the polynomial for the length l comprises computing a determinant of a Gram matrix |Bi TBi|, wherein Bi is a matrix whose i columns are i copies of a column vector of probabilities from the probability model that depend on l, to obtain the coefficient ci(l).
  12. 42. The program product according to claim 40, wherein the step of computing the coefficient ci(l) of the polynomial for the length l comprises using a recursive formula to obtain the coefficient ci(l).
  13. 43. The program product according to claim 42, wherein the recursive formula comprises
    c n + 1 ( l ) = j = 0 n ( n j ) ( - 1 ) j ( j ! ) a j + 1 c n - j ( l )
    based on the length l, coefficients cn−j(l) with c0(l)=1, and traces of powers of a second matrix M wherein an=trace(Mn).
  14. 44. The program product according to claim 43, wherein the second matrix M comprises expected values of products of pairs of probabilities according to the corresponding length l from the probability model.
  15. 45. The program product according to claim 43, wherein the second matrix M comprises
    M ij = { 1 - ( 1 - p i ) l , i = j 1 - ( 1 - p i ) l - ( 1 - p j ) l - ( 1 - p i - p j ) l , i j
    wherein pi and pj and l are from the probability model.
  16. 46. The program product according to claim 43, wherein the second matrix M comprises
    M ij = { 1 , i = j 1 - 2 ( 1 - p i ) l - 2 ( 1 - p j ) l + 4 ( 1 - p i - p j ) l , i j
    wherein pi and pj and l are from the probability model.
  17. 47. The program product according to claim 43, wherein the second matrix M comprises
    M ij = [ j 1 + + j t = l ] log j i log j j ( l j 1 j t ) p 1 j 1 p t j t
    wherein pi and pj and l are from the probability model.
  18. 48. The program product according to claim 43, wherein the second matrix M comprises
    M ij = { ( l ( l - 1 ) p i 2 + lp i ) , i = j l ( l - 1 ) p i p j , i j
    wherein pi and pj and l are from the probability model.
  19. 49. The program product according to claim 43, wherein the second matrix M comprises
    M ij = [ j 1 + + j t = l ] j i j j ( l j 1 j t ) p 1 j 1 p t j t
    wherein pi and pj and l are from the probability model.
  20. 50. The program product according to claim 43, wherein the second matrix M comprises
    M ij = { 1 - t - 1 ( t - 1 ) l , i = j t l - 2 ( t - 1 ) l + ( t - 2 ) l t l = k = 0 t - 3 ( t - 3 k ) ( t - k - 1 ) ! S ( l + 1 , t - k ) t l , i j
    wherein t and l are from the probability model and S(n,k) are the Stirling numbers of the second kind.
  21. 51. The program product according to claim 43, wherein the step of computing traces of powers of the second matrix M comprises summing j+1 powers of eigenvalues of the second matrix M to obtain the value aj+1 for use in the recursive formula.
  22. 52. A program product including a computer readable computer program encoded in a storage medium, the computer program executing an algorithm for computing an expected characteristic value of matrices created from a probability model comprising the steps of:
    providing a probability model; and
    extrapolating from the probability model an expected characteristic value of matrices created from the probability model.
  23. 53. The program product according to claim 52, wherein the characteristic value is a singular value.
  24. 54. The program product according to claim 52, wherein the characteristic value is an eigenvalue.
  25. 55. The program product according to claim 52, wherein the probability model comprises:
    a probability distribution corresponding to a set of elements;
    a probability distribution corresponding to a set of sample lengths; and
    a probability of a sample from the probability model.
  26. 56. The program product according to claim 52, wherein the step of extrapolating from the probability model comprises the steps of:
    constructing a polynomial corresponding to the probability model; and
    extrapolating from the polynomial to obtain the characteristic value.
  27. 57. The program product according to claim 56, wherein the step of extrapolating from the polynomial comprises the steps of:
    finding a root of the polynomial for the probability model; and
    extrapolating from the root of the polynomial to obtain the characteristic value.
  28. 58. The program product according to claim 57, wherein the step of extrapolating from the root of the polynomial comprises the steps of:
    multiplying the root by an expected number of samples from the probability model; and
    taking the square root of the resulting value to obtain the characteristic value.
  29. 59. The program product according to claim 56, wherein the step of constructing the polynomial comprises the steps of:
    computing one or more coefficients ci; and
    dividing each computed coefficient ci by i! to obtain the polynomial
    λ t - c 1 1 ! λ t - 1 + c 2 2 ! λ t - 2 - ± c t t ! λ 0
    wherein each of the non-computed coefficients is set to zero and t is from the probability model.
  30. 60. The program product according to claim 59, wherein the step of computing a coefficient ci of the polynomial comprises computing a probabilistically weighted value
    c i = l c i ( l ) prob ( l )
    comprising an additional coefficient ci(l) corresponding to a length l and probability prob(l) from the probability model to obtain the coefficient ci.
  31. 61. The program product according to claim 60, wherein the step of computing the coefficient ci(l) of the polynomial for the length l comprises computing a determinant of a Gram matrix |Bi TBi|, wherein Bi is a matrix whose i columns are i copies of a column vector of probabilities from the probability model that depend on l, to obtain the coefficient ci(l).
  32. 62. The program product according to claim 60, wherein the step of computing the coefficient ci(l) of the polynomial for the length l comprises using a recursive formula to obtain the coefficient ci(l).
  33. 63. The program product according to claim 62, wherein the recursive formula comprises
    c n + 1 ( l ) = j = 0 n ( n j ) ( - 1 ) j ( j ! ) a j + 1 c n - j ( l )
    based on the length l, coefficients cn-j(l) with c0(l)=1, and traces of powers of a second matrix M wherein an=trace(Mn).
  34. 64. The program product according to claim 63, wherein the second matrix M comprises expected values of products of pairs of probabilities according to the corresponding length/from the probability model.
  35. 65. The program product according to claim 63, wherein the second matrix M comprises
    M ij = { 1 - ( 1 - p i ) l , i = j 1 - ( 1 - p i ) l - ( 1 - p j ) l - ( 1 - p i - p j ) l , i j
    wherein pi and pj and l are from the probability model.
  36. 66. The program product according to claim 63, wherein the second matrix M comprises
    M ij = { 1 , i = j 1 - 2 ( 1 - p i ) l - 2 ( 1 - p j ) l + 4 ( 1 - p i - p j ) l , i j
    wherein pi and pj and l are from the probability model.
  37. 67. The program product according to claim 63, wherein the second matrix M comprises
    M ij = [ j 1 + + j t = l ] log j i log j j ( l j 1 j t ) p 1 j 1 p t j t
    wherein pi and pj and l are from the probability model.
  38. 68. The program product according to claim 63, wherein the second matrix M comprises
    M ij = { ( l ( l - 1 ) p i 2 + lp i ) , i = j l ( l - 1 ) p i p j , i j
    wherein pi and pj and l are from the probability model.
  39. 69. The program product according to claim 63, wherein the second matrix M comprises
    M ij = [ j 1 + + j t = l ] j i j j ( l j 1 j t ) p 1 j 1 p t j t
    wherein pi and pj and l are from the probability model.
  40. 70. The program product according to claim 63, wherein the second matrix M comprises
    M ij = { 1 - t - 1 ( t - 1 ) l , i = j t l - 2 ( t - 1 ) l + ( t - 2 ) l t l = k = 0 t - 3 ( t - 3 k ) ( t - k - 1 ) ! S ( l + 1 , t - k ) t l , i j
    wherein t and l are from the probability model and S(n,k) are the Stirling numbers of the second kind.
  41. 71. The program product according to claim 63, wherein the step of computing traces of powers of the second matrix M comprises summing j+1 powers of eigenvalues of the second matrix M to obtain the value aj−1 for use in the recursive formula.
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