Yun-Bin Zhao

Yunbin Zhao, Masao Fukushima

The problem of finding a rank-one solution to a system of linear matrix equations arises from many practical applications. Given a system of linear matrix equations, however, such a low-rank solution does not always exist. In this paper, we aim at developing some sufficient conditions for the existence of a rank-one solution to the system of homogeneous linear matrix equations (HLME) over the positive semidefinite cone. First, we prove that an existence condition of a rank-one solution can be established by a homotopy invariance theorem. The derived condition is closely related to the so-called $P_\emptyset$ property of the function defined by quadratic transformations. Second, we prove that the existence condition for a rank-one solution can be also established through the maximum rank of the (positive semidefinite) linear combination of given matrices. It is shown that an upper bound for the rank of the solution to a system of HLME over the positive semidefinite cone can be obtained efficiently by solving a semidefinite programming (SDP) problem. Moreover, a sufficient condition for the nonexistence of a rank-one solution to the system of HLME is also established in this paper.

Yun-Bin Zhao

Matrix rank minimization problems are gaining a plenty of recent attention in both mathematical and engineering fields. This class of problems, arising in various and across-discipline applications, is known to be NP-hard in general. In this paper, we aim at providing an approximation theory for the rank minimization problem, and prove that a rank minimization problem can be approximated to any level of accuracy via continuous optimization (especially, linear and nonlinear semidefinite programming) problems. One of the main results in this paper shows that if the feasible set of the problem has a minimum rank element with the least F-norm (i.e., Frobenius norm), then the solution of the approximation problem converges to the minimum rank solution of the original problem as the approximation parameter tends to zero. The tractability under certain conditions and convex relaxation of the approximation problem are also discussed. The methodology and results in this paper provide a new theoretical basis for the development of some efficient computational methods for solving rank minimization problems. An immediate application of this theory to the system of quadratic equations is presented in this paper. It turns out that the condition for such a system without a nonzero solution can be characterized by a rank minimization problem, and thus the proposed approximation theory can be used to establish some sufficient conditions for the system to possess only zero solution.

Yun-Bin Zhao

The uniqueness of sparsest solutions of underdetermined linear systems plays a fundamental role in the newly developed compressed sensing theory. Several new algebraic concepts, including the sub-mutual coherence, scaled mutual coherence, coherence rank, and sub-coherence rank, are introduced in this paper in order to develop new and improved sufficient conditions for the uniqueness of sparsest solutions. The coherence rank of a matrix with normalized columns is the maximum number of absolute entries in a row of its Gram matrix that are equal to the mutual coherence. The main result of this paper claims that when the coherence rank of a matrix is low, the mutual-coherence-based uniqueness conditions for the sparsest solution of a linear system can be improved. Furthermore, we prove that the Babel-function-based uniqueness can be also improved by the so-called sub-Babel function. Moreover, we show that the scaled-coherence-based uniqueness conditions can be developed, and that the right-hand-side vector $b$ of a linear system, the support overlap of solutions, the orthogonal matrix out of the singular value decomposition of a matrix, and the range property of a transposed matrix can be also integrated into the criteria for the uniqueness of the sparsest solution of an underdetermined linear system.

Yun-Bin Zhao

The Kantorovich function $(x^TAx)(x^T A^{-1} x)$, where $A$ is a positive definite matrix, is not convex in general. From matrix/convex analysis point of view, it is interesting to address the question: When is this function convex? In this paper, we investigate the convexity of this function by the condition number of its matrix. In 2-dimensional space, we prove that the Kantorovich function is convex if and only if the condition number of its matrix is bounded above by $3+2\sqrt{2}, $ and thus the convexity of the function with two variables can be completely characterized by the condition number. The upper bound `$3+2\sqrt{2} $' is turned out to be a necessary condition for the convexity of Kantorovich functions in any finite-dimensional spaces. We also point out that when the condition number of the matrix (which can be any dimensional) is less than or equal to $\sqrt{5+2\sqrt{6}}, $ the Kantorovich function is convex. Furthermore, we prove that this general sufficient convexity condition can be remarkably improved in 3-dimensional space. Our analysis shows that the convexity of the function is closely related to some modern optimization topics such as the semi-infinite linear matrix inequality or 'robust positive semi-definiteness' of symmetric matrices. In fact, our main result for 3-dimensional cases has been proved by finding an explicit solution range to some semi-infinite linear matrix inequalities.

Yunbin Zhao, Songhao Zhu, Dongsheng Wang et al.

Occluded person re-identification is one of the challenging areas of computer vision, which faces problems such as inefficient feature representation and low recognition accuracy. Convolutional neural network pays more attention to the extraction of local features, therefore it is difficult to extract features of occluded pedestrians and the effect is not so satisfied. Recently, vision transformer is introduced into the field of re-identification and achieves the most advanced results by constructing the relationship of global features between patch sequences. However, the performance of vision transformer in extracting local features is inferior to that of convolutional neural network. Therefore, we design a partial feature transformer-based person re-identification framework named PFT. The proposed PFT utilizes three modules to enhance the efficiency of vision transformer. (1) Patch full dimension enhancement module. We design a learnable tensor with the same size as patch sequences, which is full-dimensional and deeply embedded in patch sequences to enrich the diversity of training samples. (2) Fusion and reconstruction module. We extract the less important part of obtained patch sequences, and fuse them with original patch sequence to reconstruct the original patch sequences. (3) Spatial Slicing Module. We slice and group patch sequences from spatial direction, which can effectively improve the short-range correlation of patch sequences. Experimental results over occluded and holistic re-identification datasets demonstrate that the proposed PFT network achieves superior performance consistently and outperforms the state-of-the-art methods.