CN116777004B - Non-convex discriminative transfer subspace learning method incorporating distribution alignment information - Google Patents

Abstract

The invention discloses a non-convex discrimination migration subspace learning method integrated with distribution alignment information, which comprises the following steps: s1, constructing an objective function by cutting the square of the Frobenius norm; s2, carrying out iterative solution on the objective function through an IALM algorithm to obtain a projection matrix P; s3, projecting the source domain data and the target domain data into a common characteristic subspace through a projection matrix P. Compared with trace norms, the invention adopts a non-convex regularization term which is more compact to rank function approximation as a proxy model, better approximates low-rank constraint by minimizing the first k minimum singular values of a reconstruction matrix, and aligns class information of data by minimizing joint distribution difference between a source domain and a target domain, so that the source domain data can reconstruct the target domain data better, and classification performance is improved.

Description

Non-convex discriminative transfer subspace learning method incorporating distribution alignment information

Technical field

The present invention relates to the field of unsupervised technology, and more specifically, to a non-convex discriminative transfer subspace learning method that incorporates distribution alignment information.

Background technique

With the development of the big data era, in fields such as image recognition, natural language processing, and medical health, data often come from different domains and are therefore distributed differently. At the same time, massive amounts of data are often unlabeled. Therefore, how to classify these unlabeled data (i.e., target domain data) is a very important issue and has attracted the attention of many experts and scholars. Unsupervised domain adaptation provides a natural way to solve the above problems by deeply mining the information in labeled data (ie, source domain data) to promote the learning of the target classifier.

In unsupervised domain adaptation, traditional methods can be divided into two categories: distribution-based adaptation and transfer subspace learning-based. The first type of method reduces the difference between the two domains by aligning the joint distribution of the source domain and the target domain. Measuring the difference between two domains can be achieved by defining distances in different ways, among which the MMD criterion is widely adopted because of its simplicity and solid theoretical foundation. There are many methods based on distribution adaptation. Here are some classic methods. When solving the problem of different distributions in the source domain and target domain, transfer component analysis (TCA) maps the data in the two fields together into a high-dimensional regenerative kernel Hilbert space. In this space, the difference in marginal distribution between the two is minimized while maximizing their respective internal properties. On this basis, Joint Distribution Adaptation (JDA) considers both the edge distribution differences and class conditional distribution differences of the data to achieve a more refined alignment of the two domain distributions from the whole to each type. Among them, the calculation of the class conditional distribution difference requires the use of labels of the target domain data, and this is precisely our task. In response to this problem, JDA proposed the idea of pseudo tags. At the same time, considering the unreliability of pseudo-labels, an iterative update mechanism is used to reduce adverse effects. However, JDA did not make appropriate adjustments to the weights of marginal distribution differences and conditional distribution differences according to the application scenario. For example, when the data sets are more different, it means that the marginal distributions are more different; and when the data sets are similar, it means that the conditional distribution needs more attention. Therefore, if the weights of the two are set to fixed values, it will only reduce the performance of the algorithm. In response to this idea, Wang et al. proposed balanced distribution adaptation (BDA). At the same time, the author also proposed a new Weighted Balanced Distribution Adaptation (W-BDA) algorithm, which solves the class imbalance problem in migration by adaptively changing the weights of various types. In addition, Zhang et al. believe that previous methods calculate the weighted sum of the difference in marginal distributions and class conditional distributions between two domains, and a more natural measure is to directly calculate the difference in their joint probability distributions. The author also proposed that previous methods only increase the transferability between domains, but ignore the discriminability between different types of data, which may lead to a decrease in model classification performance. Therefore, joint probability distribution adaptation (JPDA) is proposed to solve this problem. One question. Wang et al. explored the essence of MMD through theoretical derivation and verified that minimizing the difference in class conditional distribution is equivalent to minimizing the inter-class distance between the source domain and the target domain, and is also equivalent to maximizing the data variance and the difference between the classes in the two domains. The inter-class distance, which intuitively illustrates the reason for the decline in feature discriminability in MMD. Through the guidance of theory, Wang et al. proposed a new discriminative MMD with two parallel strategies to alleviate the negative impact of MMD on feature discriminability.

Compared with the first type of method, the method based on migration subspace learning starts from the geometric structure characteristics of the data, making the expression more concise and effective. It can be divided into two categories: geodesic flow sampling (SGF) and geodesic flow sampling. Methods represented by the linear flow kernel method (GFK) introduce manifold learning to preserve the data structure. They believe that the domain is a point in the Grassmann manifold, connected by d intermediate points on the geodesic distance between the two to form a path. Therefore, by finding the appropriate transformation at each step, the transformation from the source domain to the target domain can be achieved. Another method considers aligning the statistical characteristics of source and target domain data to achieve knowledge transfer, such as subspace alignment (SA), subspace distribution alignment (SDA), and variance correlation alignment (CORAL). However, these methods are susceptible to interference from noise and outliers, so Shao et al. proposed low-rank transfer subspace learning (LTSL). It migrates data in the source domain and target domain into a unified generalized subspace, realizes linear reconstruction of target domain samples through source domain samples, and imposes low-rank constraints on the reconstruction matrix to ensure the global structure of the data. Considering the impact of noise during the migration process, the noise term is introduced to improve the robustness of the method. Fang et al. used least squares to fit the labels. Taking into account the adverse impact of label noise on the model, the labels of the source domain data are relaxed. This operation provides more freedom to adapt to the labels while increasing the margin between different types of data as much as possible. Discriminative transfer subspace learning (DTSL) combines the above two and believes that when reconstructing the target sample, it can achieve better results by linearly representing only a few similar source domain samples instead of all samples. Therefore, sparse constraints are imposed on the reconstruction matrix to characterize the local structure of the data. The joint feature selection and structure preserving method (FSSP) and the joint low rank representation and feature selection (JLRFS) method impose structured sparse constraints on the projection matrix, prompting the model to screen out more critical features. At the same time, the graph Laplacian term is introduced to better preserve the data structure.

However, these methods still have many limitations. First, methods based on transfer subspace learning usually impose low-rank constraints on the reconstruction matrix to preserve the global structure of the data. Since the rank minimization problem is NP-hard, many scholars approximate the rank function by using the trace norm. However, this approximation method will be significantly affected by the largest singular value, while the rank function will not, so its solution may lead to serious deviations from the optimal solution. Secondly, the traditional transfer subspace learning method only uses the characteristic information of the data to reconstruct the target sample, but ignores the category information of the data. If the target sample is linearly represented by source data with similar features but different categories, it will be difficult to classify them into pairs. In response to these problems, the present invention proposes a non-convex discriminative transfer subspace learning method that incorporates distribution alignment information to solve these problems.

Contents of the invention

The purpose of the present invention is to provide a non-convex discriminant transfer subspace learning method that incorporates distribution alignment information, so as to overcome the shortcomings of the existing technology.

In order to achieve the above objects, the technical solutions adopted by the present invention are as follows:

A non-convex discriminative transfer subspace learning method incorporating distribution alignment information for image recognition includes the following steps:

S1. Construct the objective function by truncating the square of the Frobenius norm;

S2. Use the IALM algorithm to iteratively solve the objective function to obtain the projection matrix P;

S3. Project the source domain and target domain data into a common feature subspace through the projection matrix P.

Further, the step S1 specifically includes:

S10. Linearly express the target domain data through the source domain data as:

P ^T X _t = P ^T X _s Z + E

In the formula _{, X t and}

S11. Apply low-rank constraints on Z to maintain the global structure of the data. Apply l ₁ norm on Z to make it sparse to ensure the local structure of the data. At the same time, apply sparse constraints on E. The objective function is:

In the formula, α, β, and λ are regularization parameters;

S12. Approximate the low-rank condition by truncating the square of the Frobenius norm to force the first k smallest singular values to be 0, so that the objective function is transformed into:

In the formula, k=n _s -rank(Z), is the discriminant subspace learning function term, which strengthens the discriminability of the data by fitting the source domain label in the subspace. It uses the relaxed label Y°=Y _s +B⊙M to eliminate the impact of label noise on the model, Y _s is One-hot encoding of source domain data, M is a non-negative matrix, ⊙ is the Hadamard operator, and matrix B is:

S13. Apply a structured sparse l _1,2 norm to the projection matrix, and convert the objective function in step S12 to:

In the formula, γ is the balance factor;

S14. By learning P to reduce the distance between the source domain and the target domain in the subspace, the following formula is obtained based on the distribution adaptive method:

In the formula, ω is the balance factor, are the number of samples of type c in the source domain and target domain respectively, X=[X _s ,X _t ], M ₀ , M _c are defined as follows :

S15. Use pseudo labels to calculate the formula in step S14, and iteratively refine the pseudo labels, so that The final objective function is:

Further, the step S2 specifically includes:

S20. Introduce matrices Q and J, the solution of which can be equivalently obtained by minimizing the augmented Lagrangian function to obtain the objective function:

In the formula, Y ₁ , Y ₂ , Y ₃ are Lagrange multipliers, and μ is the penalty parameter;

S21. Fix the matrices Q, Z, J, E and M, and rewrite the objective function formula of step S20 as:

By calculating the partial derivative of the objective function in step S21 and setting it to 0, the closed-form solution of matrix P can be obtained:

In the formula, G ₁ =Y _s +B⊙M, G ₂ =X _t -X _s Z;

S22. Fixed matrices P, Z, J, E and M, and the objective function of updating Q is:

in Using the contraction operator, the solution of Q can be expressed as:

In the formula, [A] _:,i represents the i-th column of matrix A;

S23. Fixed matrices P, Q, J, E and M, and the objective function of updating Z is:

because The solution of is NP-hard, based on Ky Fan theory, which is equivalent to The objective function can be rewritten as:

Calculate F and Z iteratively until the above objective function converges. After obtaining the optimal solution of Z, update step S24 instead of just iterating F and Z once. Let the partial derivative of the above formula be 0 and obtain the closed-form solution of Z as follows :

in, When calculating F, the following formula is optimized:

The optimal solution of F is formed by k left singular vectors corresponding to the k minimum singular values of Z, that is, U ₂ , where U is the left singular vector matrix obtained by Z through SVD decomposition, let U=[U ₁ ,U ₂ ] , Since only FF ^T is needed to calculate the closed-form solution of Z, it is calculated by the following formula:

S24. Fixed matrices P, Q, Z, E and M, and update the objective function of J as:

Using the contraction operator, the iterative solution of J is:

In the formula, shrink(x,c)=sign max(|x|-c,0).

S25. Fixed matrices P, Q, Z, J and M, and update the objective function of E as:

Using the contraction operator, the iterative solution of E is:

S26. Fixed matrices P, Q, Z, J and E, and update the objective function of M as

Let R=P ^T X _s -Y _s , the above objective function is decomposed into d×n _s sub-optimization problems Among them, d and n _s are the number of rows and columns of the non-negative relaxed label matrix M respectively. The optimal solution is M _ij =max (R _ij B _ij ,0). The closed-form solution of the above objective function is:

M ^* =max(R⊙B,0)

S27. Update Lagrange multipliers Y ₁ , Y ₂ , Y ₃ and penalty factor μ:

In the formula, ρ is the learning rate, and μ _max is the maximum value that μ can take.

Compared with the existing technology, the advantage of the present invention is that: the present invention uses a non-convex regular term that is more compact in approximating the rank function than the trace norm as a proxy model, which reconstructs the matrix by minimizing The first k minimum singular values are used to better approximate the low-rank constraint, and the category information of the data is aligned by minimizing the joint distribution difference between the source domain and the target domain, so that the source domain data can better reconstruct the target domain data, improving Classification performance.

Description of drawings

In order to explain the embodiments of the present invention or the technical solutions in the prior art more clearly, the drawings needed to be used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings in the following description are only These are some embodiments of the present invention. For those of ordinary skill in the art, other drawings can be obtained based on these drawings without exerting creative efforts.

Figure 1 is an iterative optimization solution diagram of the non-convex discriminant transfer subspace learning method of the present invention that incorporates distribution alignment information.

Detailed ways

The preferred embodiments of the present invention are described in detail below in conjunction with the accompanying drawings, so that the advantages and features of the present invention can be more easily understood by those skilled in the art, and the protection scope of the present invention can be more clearly defined.

Traditional transfer subspace learning methods usually use trace norm to approximate low rank. However, it is too relaxed, causing its solution to seriously deviate from the ideal solution. In addition, traditional methods ignore the category information of aligned data, resulting in reduced classification performance. Therefore, this embodiment proposes a non-convex discriminative transfer subspace learning method that incorporates distribution alignment information to solve these problems. Specifically, the goal of this embodiment is to learn the projection matrix P to project the source domain and the target domain into a common subspace.

Referring to Figure 1, this embodiment discloses a non-convex discriminant transfer subspace learning method that incorporates distribution alignment information for image recognition, including the following steps:

Step S1: Construct the objective function by truncating the square of the Frobenius norm, specifically including:

Step S10. Since the two domain data are in the same manifold, the target domain data can be expressed linearly through the source domain data. The original formula is written as:

P ^T X _t = P ^T X _s Z + E (1)

In the formula _{, X t and}

Step S11: Each target sample can be linearly represented by a combination of its source domain neighbor samples. Therefore, a low-rank constraint is imposed on the reconstruction matrix Z to maintain this property and protect the global structure of the data. At the same time, the reconstruction of each target sample only requires the participation of a few source domain samples. The l ₁ norm is applied on Z to make it sparse to ensure the local structure of the data. The objective function is:

In the formula, α, β, and λ are regularization parameters.

Step S12. Since the trace norm is the most compact convex approximation of the rank function on the unit sphere, many experts and scholars use it as a surrogate model to approximate low-rank conditions. However, according to the definition of the trace norm, it will change drastically with the significant change of the maximum singular value of the matrix, but the rank of the matrix will not change. Therefore, this relaxation method of trace norm will cause its solution to seriously deviate from the ideal solution. This embodiment approximates the low-rank condition by truncating the square of the Frobenius norm, causing the first k smallest singular values to be 0, eliminating the impact of other larger singular values on the surrogate model, and converting the objective function (2) into:

In the formula, k=n _s -rank(Z), is the discriminant subspace learning function term, which strengthens the discriminability of the data by fitting the source domain label in the subspace. It uses the relaxed label Y°=Y _s +B⊙M to eliminate the impact of label noise on the model, Y _s is One-hot encoding of source domain data, M is a non-negative matrix, ⊙ is the Hadamard operator, and matrix B is:

Step S13. In addition, the selected features may be redundant, so it is necessary to apply a structured sparse l _1,2 norm on the projection matrix to convert the objective function in step S12 into:

In the formula, γ is the balance factor.

Step S14. However, traditional methods and the above formulas only use the characteristic information of the data to align the subspaces of the source domain and the target domain, ignoring the impact of data category information on the model. Therefore, this embodiment uses different categories of semantic information to reduce the distribution difference between the two domains, thereby improving the classification performance of the model. Specifically, this embodiment uses learning P to reduce the distance between the two domains in the subspace, and obtains the following formula based on the distribution adaptive method:

In the formula, ω is the balance factor, are the number of samples of type c in the source domain and target domain respectively, X=[X _s ,X _t ], M ₀ , M _c are defined as follows:

Step S15. However, when calculating formula (5), since the target domain has no label, it is impossible to know whether the j-th sample in the target domain belongs to the c-th category, that is, unknown. Therefore, pseudo labels are used for calculation. And because the reliability of pseudo-labels is low, the pseudo-labels are iteratively refined, so that/> The final objective function is:

Step S2: Iteratively solve the objective function through the IALM algorithm to obtain the projection matrix P.

Specifically, this embodiment designs the IALM algorithm to effectively solve the objective function (6). The step S2 specifically includes:

Step S20, introduce matrices Q and J, the solution of which can be obtained equivalently by minimizing the augmented Lagrangian function (7):

In the formula, Y ₁ , Y ₂ , Y ₃ are Lagrange multipliers, and μ is the penalty parameter;

Step S21, fix the matrices Q, Z, J, E and M, and rewrite the objective function formula of step S20 as:

By calculating the partial derivative of the objective function (8) and setting it to 0, the closed-form solution of the matrix P can be obtained:

In the formula, G ₁ =Y _s +B⊙M, G ₂ =X _t -X _s Z;

Step S22, fix the matrices P, Z, J, E and M, and update the objective function of Q as:

in Using the contraction operator, the solution of Q can be expressed as:

In the formula, [A] _:,i represents the i-th column of matrix A;

Step S23, fix the matrices P, Q, J, E and M, and update the objective function of Z as:

because The solution of is NP-hard, based on Ky Fan theory, which is equivalent to The objective function can be rewritten as:

Calculate F and Z iteratively until the above objective function converges. After obtaining the optimal solution of Z, update step S24 instead of just iterating F and Z once. Let the partial derivative of formula (13) be 0 to obtain the closed formula of Z. The solution is as follows:

in, When calculating F, the following formula is optimized:

The optimal solution of F is formed by k left singular vectors corresponding to the k minimum singular values of Z, that is, U ₂ , where U is the left singular vector matrix obtained by Z through SVD decomposition, let U=[U ₁ ,U ₂ ] , Since only FF ^T is needed to calculate the closed-form solution of Z, it is calculated by the following formula:

Step S24, fix the matrices P, Q, Z, E and M, and update the objective function of J as:

Using the contraction operator, the iterative solution of J is:

In the formula, shrink(x,c)=sign max(|x|-c,0).

Step S25, fix the matrices P, Q, Z, J and M, and update the objective function of E as:

Using the contraction operator, the iterative solution of E is:

Step S26, fix the matrices P, Q, Z, J and E, and update the objective function of M as

Let R=P ^T X _s -Y _s , the above objective function is decomposed into d×n _s sub-optimization problems Among them, d and n _s are the number of rows and columns of the non-negative relaxed label matrix M respectively. The optimal solution is M _ij =max (R _ij B _ij ,0). The closed-form solution of the above objective function is:

M ^* =max(R⊙B,0) (21)

Step S27: Update Lagrange multipliers Y ₁ , Y ₂ , Y ₃ and penalty factor μ:

In the formula, ρ is the learning rate, and μ _max is the maximum value that μ can take.

To sum up, the various optimization steps of the model are summarized in Table 1 below:

Step S3: Project the source domain and target domain data into a common feature subspace through the projection matrix P.

This embodiment also improves the above algorithm. Another variation is used when approximating the rank function It is constructed by truncating the trace norm. Compared with the approximate method in the previous article/> Variants/> The formula definition is closer to the rank function, so the approximation effect is better. The objective function can be obtained as follows:

The analysis shows that changing the surrogate model with low-rank conditions only affects the calculation of the reconstruction matrix Z. Therefore, when other variables are fixed and Z is updated, the above objective function changes to:

Similarly, It is NP-hard to solve. Fortunately, this can be converted equivalently to

where r=rank(Z). When optimizing the trace norm, the present invention converts it into Tr(Z ^T DZ) and solves it iteratively. in represents the current optimal solution. Therefore, the objective function (24) can be transformed into

When fixing matrices F and G, update Z. The closed form solution of Z can be obtained as follows:

When Z is fixed, the optimal solutions of F and G are respectively composed of the left and right singular vectors corresponding to the first r largest singular values of Z.

The present invention uses a non-convex regular term that is more compact in approximating the rank function than the trace norm as a surrogate model. It can better approximate low rank by minimizing the first k smallest singular values of the reconstruction matrix. Constraints are used to align the category information of the data by minimizing the joint distribution difference between the source domain and the target domain, so that the source domain data can better reconstruct the target domain data and improve classification performance.

Although the embodiments of the present invention have been described in conjunction with the accompanying drawings, the patent owner may make various variations or modifications within the scope of the appended claims, as long as they do not exceed the protection scope described in the claims of the present invention. within the protection scope of the present invention.

Claims (1)

1. The non-convex discrimination migration subspace learning method integrated with the distribution alignment information is used for image recognition and is characterized by comprising the following steps of:

s1, constructing an objective function by cutting the square of the Frobenius norm;

s2, carrying out iterative solution on the objective function through an IALM algorithm to obtain a projection matrix P;

s3, projecting the source domain data and the target domain data into a common feature subspace through a projection matrix P;

the step S1 specifically includes:

s10, linearly representing the target domain data by the source domain data as follows:

P ^T X _t ＝P ^T X _s Z+E

wherein X is _t And X _s Feature matrices of the target domain data and the source domain data respectively; wherein n is _t And n _s The data quantity of the two domains respectively; z is a reconstruction matrix, E is a noise matrix;

s11, applying low-rank constraint on Z to maintain global structure of data, and applying l ₁ The norm promotes the sparsity on Z to ensure the local structure of the data, and simultaneously applies sparsity constraint to E to obtain an objective function as follows:

wherein alpha, beta, lambda are regularization parameters;

s12, the square of the Frobenius norm is truncated to promote the first k minimum singular values to be 0 to approximate a low-rank condition, and an objective function is converted into:

where k=n _s -rank(Z)，Is a discriminant subspace learning function term that enhances the discriminant of data by fitting source domain labels in subspaces, using the relaxed labels y+=y _s +B.sup.M to eliminate the influence of tag noise on the model, Y _s For the single thermal encoding of source domain data, M is a non-negative matrix, and by Hadamard operator, matrix B is:

s13, applying structured sparsity on the projection matrix _1,2 The norm converts the objective function in step S12 into:

wherein, gamma is a balance factor;

s14, reducing the distance between a source domain and a target domain in the subspace by learning P, and obtaining the following formula based on a distribution self-adaption method:

where ω is the balance factor,the number of class c samples in the source domain and the target domain, x= [ X ] _s ,X _t ]，M ₀ ,M _c The definition is as follows:

s15, calculating the formula in the step S14 by adopting the pseudo tag, and carrying out iterative refinement on the pseudo tag to enable the pseudo tag to be madeThe final objective function is:

the step S2 specifically includes:

s20, introducing matrixes Q and J, wherein the solution can be equivalently obtained into an objective function by minimizing the augmented Lagrangian function:

wherein Y is ₁ ,Y ₂ ,Y ₃ Is Lagrangian multiplier, μ is penalty parameter;

initializing: p=p ₀ ,Q＝0，M＝1,J＝Z＝0,E＝0,Y ₁ ＝0,Y ₂ ＝0,Y ₃ ＝0,μ ₀ ＝0.05,μ _max ＝10 ⁶ ,ρ＝1.1,ε＝10 ^-6 Repeating the following steps until the objective function meets the convergence condition;

s21, fixing matrices Q, Z, J, E and M, and rewriting the objective function formula of step S20 to:

by calculating the partial derivative of the objective function in step S21 to make it 0, a closed-form solution of the matrix P can be obtained:

wherein G is ₁ ＝Y _s +B⊙M，G ₂ ＝X _t -X _s Z；

S22, fixing matrices P, Z, J, E and M, updating the objective function of Q as:

wherein the method comprises the steps ofWith a contraction operator, the solution for Q can be expressed as:

wherein [ A ]] _:,i Representing the ith column of matrix a;

s23, fixing matrices P, Q, J, E and M, updating the objective function of Z as:

due toIs NP-difficult, which is equivalent to +.>The objective function may be rewritten as:

and (3) carrying out iterative computation on F and Z until the objective function converges to obtain an optimal solution of Z, and then updating in the step S24, instead of only iterating F and Z once, enabling the partial derivative of the formula to be 0, so as to obtain a closed solution of Z as follows:

wherein,in calculating F, the following formula is optimized:

the optimal solution of F is formed by k left singular vectors corresponding to k minimum singular values of Z, namely U ₂ Wherein U is a left singular vector matrix obtained by SVD decomposition of Z, and U= [ U ₁ ,U ₂ ]， Since only FF is needed in computing the closed-form solution of Z ^T Calculated by the following formula:

s24, fixing matrices P, Q, Z, E and M, updating the objective function of J as:

the iterative solution of J is:

where shrnk (x, c) =sign max (|x| -c, 0);

s25, fixing the matrices P, Q, Z, J and M, updating the objective function of E as:

the iterative solution of E is:

s26, fixing matrices P, Q, Z, J and E, updating the objective function of M to be

Let r=p ^T X _s -Y _s The objective function is decomposed into d×n _s Sub-optimization problemWherein d, n _s The number of rows and columns of the non-negative loose label matrix M are respectively, and the optimal solution is M _ij ＝max(R _ij B _ij 0), the closed-form solution of the objective function is:

M ^* ＝max(R⊙B,0)

s27, updating Lagrangian multiplier Y ₁ ,Y ₂ ,Y ₃ Penalty factor μ:

wherein ρ is learning rate, μ _max Is the maximum value that mu can take.