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Register a new userLearning Log-Determinant Divergences for Positive Definite Matrices
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Added by
Anoop Cherian
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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Representations in the form of Symmetric Positive Definite (SPD) matrices have been popularized in a variety of visual learning applications due to their demonstrated ability to capture rich second-order statistics of visual data. There exist several similarity measures for comparing SPD matrices with documented benefits. However, selecting an appropriate measure for a given problem remains a challenge and in most cases, is the result of a trial-and-error process. In this paper, we propose to learn similarity measures in a data-driven manner. To this end, we capitalize on the alphabeta-log-det divergence, which is a meta-divergence parametrized by scalars alpha and beta, subsuming a wide family of popular information divergences on SPD matrices for distinct and discrete values of these parameters. Our key idea is to cast these parameters in a continuum and learn them from data. We systematically extend this idea to learn vector-valued parameters, thereby increasing the expressiveness of the underlying non-linear measure. We conjoin the divergence learning problem with several standard tasks in machine learning, including supervised discriminative dictionary learning and unsupervised SPD matrix clustering. We present Riemannian gradient descent schemes for optimizing our formulations efficiently, and show the usefulness of our method on eight standard computer vision tasks.
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In this paper, we develop a new classification method for manifold-valued data in the framework of probabilistic learning vector quantization. In many classification scenarios, the data can be naturally represented by symmetric positive definite matrices, which are inherently points that live on a curved Riemannian manifold. Due to the non-Euclidean geometry of Riemannian manifolds, traditional Euclidean machine learning algorithms yield poor results on such data. In this paper, we generalize the probabilistic learning vector quantization algorithm for data points living on the manifold of symmetric positive definite matrices equipped with Riemannian natural metric (affine-invariant metric). By exploiting the induced Riemannian distance, we derive the probabilistic learning Riemannian space quantization algorithm, obtaining the learning rule through Riemannian gradient descent. Empirical investigations on synthetic data, image data , and motor imagery EEG data demonstrate the superior performance of the proposed method.
We study metric properties of symmetric divergences on Hermitian positive definite matrices. In particular, we prove that the square root of these divergences is a distance metric. As a corollary we obtain a proof of the metric property for Quantum Jensen-Shannon-(Tsallis) divergences (parameterized by $alphain [0,2]$), which in turn (for $alpha=1$) yields a proof of the metric property of the Quantum Jensen-Shannon divergence that was conjectured by Lamberti emph{et al.} a decade ago (emph{Metric character of the quantum Jensen-Shannon divergence}, Phy. Rev. A, textbf{79}, (2008).) A somewhat more intricate argument also establishes metric properties of Jensen-Renyi divergences (for $alpha in (0,1)$), and outlines a technique that may be of independent interest.
In a number of disciplines, the data (e.g., graphs, manifolds) to be analyzed are non-Euclidean in nature. Geometric deep learning corresponds to techniques that generalize deep neural network models to such non-Euclidean spaces. Several recent papers have shown how convolutional neural networks (CNNs) can be extended to learn with graph-based data. In this work, we study the setting where the data (or measurements) are ordered, longitudinal or temporal in nature and live on a Riemannian manifold -- this setting is common in a variety of problems in statistical machine learning, vision and medical imaging. We show how recurrent statistical recurrent network models can be defined in such spaces. We give an efficient algorithm and conduct a rigorous analysis of its statistical properties. We perform extensive numerical experiments demonstrating competitive performance with state of the art methods but with significantly less number of parameters. We also show applications to a statistical analysis task in brain imaging, a regime where deep neural network models have only been utilized in limited ways.
The Riemannian metric on the manifold of positive definite matrices is defined by a kernel function $phi$ in the form $K_D^phi(H,K)=sum_{i,j}phi(lambda_i,lambda_j)^{-1} Tr P_iHP_jK$ when $sum_ilambda_iP_i$ is the spectral decomposition of the foot point $D$ and the Hermitian matrices $H,K$ are tangent vectors. For such kernel metrics the tangent space has an orthogonal decomposition. The pull-back of a kernel metric under a mapping $Dmapsto G(D)$ is a kernel metric as well. Several Riemannian geometries of the literature are particular cases, for example, the Fisher-Rao metric for multivariate Gaussian distributions and the quantum Fisher information. In the paper the case $phi(x,y)=M(x,y)^theta$ is mostly studied when $M(x,y)$ is a mean of the positive numbers $x$ and $y$. There are results about the geodesic curves and geodesic distances. The geometric mean, the logarithmic mean and the root mean are important cases.
Let $mathbb{C}^{ntimes n}$ be the set of all $n times n$ complex matrices. For any Hermitian positive semi-definite matrices $A$ and $B$ in $mathbb{C}^{ntimes n}$, their new common upper bound less than $A+B-A:B$ is constructed, where $(A+B)^dag$ denotes the Moore-Penrose inverse of $A+B$, and $A:B=A(A+B)^dag B$ is the parallel sum of $A$ and $B$. A factorization formula for $(A+X):(B+Y)-A:B-X:Y$ is derived, where $X,Yinmathbb{C}^{ntimes n}$ are any Hermitian positive semi-definite perturbations of $A$ and $B$, respectively. Based on the derived factorization formula and the constructed common upper bound of $X$ and $Y$, some new and sharp norm upper bounds of $(A+X):(B+Y)-A:B$ are provided. Numerical examples are also provided to illustrate the sharpness of the obtained norm upper bounds.
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