CN105550645B

CN105550645B - A kind of least squared classified method in product Grassmann manifold

Info

Publication number: CN105550645B
Application number: CN201510901535.0A
Authority: CN
Inventors: 尹宝才; 王玉萍; 王立春; 孔德慧
Original assignee: Beijing University of Technology
Current assignee: Beijing University of Technology
Priority date: 2015-12-08
Filing date: 2015-12-08
Publication date: 2019-02-12
Anticipated expiration: 2035-12-08
Also published as: CN105550645A

Abstract

The present invention discloses a kind of least squared classified method in product Grassmann manifold, has closing solution, can be improved the accuracy of identification.The method comprising the steps of: (1) carrying out product Grassmann manifold to video indicates；(2) least square model is established in Grassmann manifold and is solved；(3) least squared classified, and output category result are carried out.

Description

A kind of least squared classified method in product Grassmann manifold

Technical field

The invention belongs to the technical fields of pattern-recognition, more particularly to the minimum in a kind of product Grassmann manifold Two multiply classification method.

Background technique

In recent years, linear subspaces method has critically important application, such as target identification, face in terms of computer vision Identification, human body tracking etc..Linear subspaces can reduce the inherent geometry knot for calculating cost and capable of preferably portraying data itself Structure.Grassmann manifold is a kind of linear subspaces with nonlinear organization, and video data is passed through high-order SVD breakdown The point being shown as in product Grassmann manifold has significant effect in gesture identification.For identifying problem, in addition to find Character representation except, the classification method of robust also functions to vital effect to recognition correct rate.

As most simple effective method in statistical analysis, the research in manifold space also obtains least square method The concern of many scholars.Lui gives the non-linear least square method in Grassmann manifold by means of kernel function, uses Geodesic curve distance is measured.However Weighted Karcher Mean algorithm is used in this method Solve problems, it is one Iterative algorithm, what is obtained is an approximate solution.

Summary of the invention

Technology of the invention solves the problems, such as: overcoming the deficiencies of the prior art and provide in a kind of product Grassmann manifold Least squared classified method, have closing solution, can be improved the accuracy of identification.

The technical solution of the invention is as follows: the least squared classified method in this product Grassmann manifold, the party Method the following steps are included:

(1) carrying out product Grassmann manifold to video indicates；

(2) least square model is established in Grassmann manifold and is solved；

(3) least squared classified, and output category result are carried out.

The present invention carries out degree of error by the way that the point in Grassmann manifold to be equidistantly embedded into Symmetric matrix again Amount can be improved the accuracy of identification so having closing solution.

Detailed description of the invention

Fig. 1 shows method frame figure of the invention.

Specific embodiment

As shown in Figure 1, the least squared classified method in this product Grassmann manifold, this method includes following step It is rapid:

(1) carrying out product Grassmann manifold to video indicates；

(2) least square model is established in Grassmann manifold and is solved；

(3) least squared classified, and output category result are carried out.

Preferably, in the step (1) representation of video shot at tensor formWherein I₁, I₂, I₃Respectively Indicate height, the width, length of video；Variation under each mode can be obtained by high-order SVD,, whereinIt is core tensor, V⁽¹⁾, V⁽²⁾, V⁽³⁾It is the factor matrix under each mode respectively, and each V^(k)It is tall and thin orthogonal matrix, is in Stiefel manifold Point, then span (V^(k)) be Grassmann manifold on point, (span (V⁽¹⁾), span (V⁽²⁾), span (V⁽³⁾)) it is product Point in Grassmann manifold.

Preferably, least squared classified in the step (2)

The solution for optimizing formula (1) is formula (3)

y^*=2 (K (D)+K (D)^T)^-1K (X, D) (3)

Corresponding error is

Preferably, sample (X, Y, Z) is defined as about the residual error of kth class in the step (3)

WhereinIt is the solution of formula (1) on each submanifold, final classification respectively As a result by k^*=arg min_kε^kIt determines.

Above method is specifically described below.

1. the product Grassmann manifold of video indicates

Video can be expressed as the form of tensor such as high dimensional dataWherein I₁, I₂, I₃Table respectively Show the height, width, length of video.Variation under each mode can be obtained by high-order SVD, i.e.,WhereinIt is core tensor, V⁽¹⁾, V⁽²⁾, V⁽³⁾It is the factor matrix under each mode respectively, and each V^(k)It is tall and thin orthogonal matrix, therefore can regards as Point in Stiefel manifold, then span (V^(k)) be Grassmann manifold on point.Therefore (span (V⁽¹⁾), span (V⁽²⁾), span (V⁽³⁾)) be product Grassmann manifold on point.

Embedded least square method and solution in 2.Grassmann manifold

Least square technology is a most simple effective method in statistical analysis.In theorem in Euclid space, parameterCan by minimize residual error R (β)=| | y-A β | |²It obtains, whereinFor training set,For regressand value.Estimation parameter have display solution shaped likeCorresponding to error at this time is | | y-A (A^TA)^-1A^Ty||²。

It enablesThe training set for being N for sample size, whereinIndicate Grassmann Manifold,It is fitting parameter,It is input sample.Utilize projection mapping

Point in Grassmann manifold is embedded into Symmetric matrix, wherein Sym (d) indicates Symmetric matrix.This The distance of upper two point X and Y in the space sample Grassmann can be defined with the distance of embedded space, i.e.,And the geodesic curve distance defined in the distance and Grassmann manifold is Valence.Such distance definition is also convenient for subsequent solution.Similar to the theorem in Euclid space principle of least square, embedded minimum two are provided Multiply to solve following optimization problems,

Wherein y_jIt is j-th of element of vector y.

It is described below and how to solve above-mentioned optimization problem.Have

Definition

Therefore model (1) becomes

min_y{y^TK(D)y-2y^TK (X, D) } (2)

To (2) about y derivation, and derivative is enabled to be equal to 0, had

(K(D)+K(D)^T) y-2K (X, D)=0

So the solution of optimization problem (1) is

y^*=2 (K (D)+K (D)^T)^-1K (X, D) (3)

Therefore corresponding error is

3. the principle of classification based on embedded least square method

3 factorial Grassmann manifolds of corresponding video data are discussed belowWherein × indicate cartesian product.Assuming that training set shares M class, kth class is remembered Training set isWherein N_kFor number of samples.Target is to infer test sampleWhich kind of belongs to.

Sample (X, Y, Z) is defined as about the residual error of kth class

WhereinIt is the solution of each submanifold upper returning problem (1) respectively.Finally Classification results are determined by following formula

k^*=arg min_kε^k

Experimental verification has been carried out to above-mentioned model, and has achieved apparent effect.In an experiment, the hand of Cambridge University is selected Gesture database, it includes 900 videos, and totally 9 different gestures, are divided into 5 set according to different illumination conditions.Set 5 generally as training set, and set 1-4 is as test set.In experiment, by original series be converted into gray level image and be sized to 14×32×23.Current high-caliber 2 kinds of recognition methods and the method for the present invention are listed in table 1 respectively on four test sets Correct recognition rata.The method of the present invention correct recognition rata is apparently higher than the discrimination of the PM method of Lui, square with Harandi et al. The best result kgLLC of method is on close level.The description of test the method for the present invention is simple and effective.

Table 1

The above is only presently preferred embodiments of the present invention, is not intended to limit the present invention in any form, it is all according to According to technical spirit any simple modification, equivalent change and modification to the above embodiments of the invention, still belong to the present invention The protection scope of technical solution.

Claims

1. a kind of least squared classified method in product Grassmann manifold, it is characterised in that: this method includes following step It is rapid:

(1) carrying out product Grassmann manifold to video indicates；

(2) least square model is established in Grassmann manifold and is solved；

(3) least squared classified, and output category result are carried out；

In the step (1) representation of video shot at tensor formWherein I₁, I₂, I₃Respectively indicate video High, wide, length；Variation under each mode can decompose to obtain by high-order SVD,Wherein

It is core tensor, V⁽¹⁾, V⁽²⁾, V⁽³⁾It is the factor square under each mode respectively Battle array, and each V^(k)It is tall and thin orthogonal matrix, is the point in Stiefel manifold, then span (V^(k)) it is that Grassmann flows Point in shape, (span (V⁽¹⁾), span (V⁽²⁾), span (V⁽³⁾)) be product Grassmann manifold on point.

2. the least squared classified method in product Grassmann manifold according to claim 1, it is characterised in that: institute State least squared classified in step (2)

WhereinIt is input sample,Indicate Grassmann manifold,The training set for being N for sample size,It is fitting parameter, | | | |_F Indicate Frobenius norm,

The solution for optimizing formula (1) is formula (3)

y^*=2 (K (D)+K (D)^T)^-1K (X, D) (3)

Corresponding error is

WhereinFor optimal solution, and

3. the least squared classified method in product Grassmann manifold according to claim 2, it is characterised in that: institute State sample in step (3)

Residual error about kth class is defined as

WhereinFor 3 factorial Grassmann manifoldsOn kth class training set, N_kFor number of samples,It is the solution of formula (1) on each submanifold respectively, final classification result is by k^*=arg min_kε^kIt determines.