Hông Vân Lê, Hà Quang Minh, Frederic Protin et al.
We present and discuss applications of the category of probabilistic morphisms, initially developed in \cite{Le2023}, as well as some geometric methods to several classes of problems in statistical, machine and manifold learning which shall be, along with many other topics, considered in depth in the forthcoming book \cite{LMPT2024}.
Ha Quang Minh
In this work we study two Riemannian distances between infinite-dimensional positive definite Hilbert-Schmidt operators, namely affine-invariant Riemannian and Log-Hilbert-Schmidt distances, in the context of covariance operators associated with functional stochastic processes, in particular Gaussian processes. Our first main results show that both distances converge in the Hilbert-Schmidt norm. Using concentration results for Hilbert space-valued random variables, we then show that both distances can be consistently and efficiently estimated from (i) sample covariance operators, (ii) finite, normalized covariance matrices, and (iii) finite samples generated by the given processes, all with dimension-independent convergence. Our theoretical analysis exploits extensively the methodology of reproducing kernel Hilbert space (RKHS) covariance and cross-covariance operators. The theoretical formulation is illustrated with numerical experiments on covariance operators of Gaussian processes.
Minh Ha Quang
This work studies the entropic regularization formulation of the 2-Wasserstein distance on an infinite-dimensional Hilbert space, in particular for the Gaussian setting. We first present the Minimum Mutual Information property, namely the joint measures of two Gaussian measures on Hilbert space with the smallest mutual information are joint Gaussian measures. This is the infinite-dimensional generalization of the Maximum Entropy property of Gaussian densities on Euclidean space. We then give closed form formulas for the optimal entropic transport plan, entropic 2-Wasserstein distance, and Sinkhorn divergence between two Gaussian measures on a Hilbert space, along with the fixed point equations for the barycenter of a set of Gaussian measures. Our formulations fully exploit the regularization aspect of the entropic formulation and are valid both in singular and nonsingular settings. In the infinite-dimensional setting, both the entropic 2-Wasserstein distance and Sinkhorn divergence are Fréchet differentiable, in contrast to the exact 2-Wasserstein distance, which is not differentiable. Our Sinkhorn barycenter equation is new and always has a unique solution. In contrast, the finite-dimensional barycenter equation for the entropic 2-Wasserstein distance fails to generalize to the Hilbert space setting. In the setting of reproducing kernel Hilbert spaces (RKHS), our distance formulas are given explicitly in terms of the corresponding kernel Gram matrices, providing an interpolation between the kernel Maximum Mean Discrepancy (MMD) and the kernel 2-Wasserstein distance.
Anton Mallasto, Augusto Gerolin, Hà Quang Minh
Gaussian distributions are plentiful in applications dealing in uncertainty quantification and diffusivity. They furthermore stand as important special cases for frameworks providing geometries for probability measures, as the resulting geometry on Gaussians is often expressible in closed-form under the frameworks. In this work, we study the Gaussian geometry under the entropy-regularized 2-Wasserstein distance, by providing closed-form solutions for the distance and interpolations between elements. Furthermore, we provide a fixed-point characterization of a population barycenter when restricted to the manifold of Gaussians, which allows computations through the fixed-point iteration algorithm. As a consequence, the results yield closed-form expressions for the 2-Sinkhorn divergence. As the geometries change by varying the regularization magnitude, we study the limiting cases of vanishing and infinite magnitudes, reconfirming well-known results on the limits of the Sinkhorn divergence. Finally, we illustrate the resulting geometries with a numerical study.
Hà Quang Minh, Marco San Biagio, Loris Bazzani et al.
This paper presents a novel framework for visual object recognition using infinite-dimensional covariance operators of input features in the paradigm of kernel methods on infinite-dimensional Riemannian manifolds. Our formulation provides in particular a rich representation of image features by exploiting their non-linear correlations. Theoretically, we provide a finite-dimensional approximation of the Log-Hilbert-Schmidt (Log-HS) distance between covariance operators that is scalable to large datasets, while maintaining an effective discriminating capability. This allows us to efficiently approximate any continuous shift-invariant kernel defined using the Log-HS distance. At the same time, we prove that the Log-HS inner product between covariance operators is only approximable by its finite-dimensional counterpart in a very limited scenario. Consequently, kernels defined using the Log-HS inner product, such as polynomial kernels, are not scalable in the same way as shift-invariant kernels. Computationally, we apply the approximate Log-HS distance formulation to covariance operators of both handcrafted and convolutional features, exploiting both the expressiveness of these features and the power of the covariance representation. Empirically, we tested our framework on the task of image classification on twelve challenging datasets. In almost all cases, the results obtained outperform other state of the art methods, demonstrating the competitiveness and potential of our framework.
Ha Quang Minh
This paper presents a framework for computing random operator-valued feature maps for operator-valued positive definite kernels. This is a generalization of the random Fourier features for scalar-valued kernels to the operator-valued case. Our general setting is that of operator-valued kernels corresponding to RKHS of functions with values in a Hilbert space. We show that in general, for a given kernel, there are potentially infinitely many random feature maps, which can be bounded or unbounded. Most importantly, given a kernel, we present a general, closed form formula for computing a corresponding probability measure, which is required for the construction of the Fourier features, and which, unlike the scalar case, is not uniquely and automatically determined by the kernel. We also show that, under appropriate conditions, random bounded feature maps can always be computed. Furthermore, we show the uniform convergence, under the Hilbert-Schmidt norm, of the resulting approximate kernel to the exact kernel on any compact subset of Euclidean space. Our convergence requires differentiable kernels, an improvement over the twice-differentiability requirement in previous work in the scalar setting. We then show how operator-valued feature maps and their approximations can be employed in a general vector-valued learning framework. The mathematical formulation is illustrated by numerical examples on matrix-valued kernels.
Vikas Sindhwani, Ha Quang Minh, Aurelie Lozano
We propose a general matrix-valued multiple kernel learning framework for high-dimensional nonlinear multivariate regression problems. This framework allows a broad class of mixed norm regularizers, including those that induce sparsity, to be imposed on a dictionary of vector-valued Reproducing Kernel Hilbert Spaces. We develop a highly scalable and eigendecomposition-free algorithm that orchestrates two inexact solvers for simultaneously learning both the input and output components of separable matrix-valued kernels. As a key application enabled by our framework, we show how high-dimensional causal inference tasks can be naturally cast as sparse function estimation problems, leading to novel nonlinear extensions of a class of Graphical Granger Causality techniques. Our algorithmic developments and extensive empirical studies are complemented by theoretical analyses in terms of Rademacher generalization bounds.
Ha Quang Minh, Loris Bazzani, Vittorio Murino
This paper presents a general vector-valued reproducing kernel Hilbert spaces (RKHS) framework for the problem of learning an unknown functional dependency between a structured input space and a structured output space. Our formulation encompasses both Vector-valued Manifold Regularization and Co-regularized Multi-view Learning, providing in particular a unifying framework linking these two important learning approaches. In the case of the least square loss function, we provide a closed form solution, which is obtained by solving a system of linear equations. In the case of Support Vector Machine (SVM) classification, our formulation generalizes in particular both the binary Laplacian SVM to the multi-class, multi-view settings and the multi-class Simplex Cone SVM to the semi-supervised, multi-view settings. The solution is obtained by solving a single quadratic optimization problem, as in standard SVM, via the Sequential Minimal Optimization (SMO) approach. Empirical results obtained on the task of object recognition, using several challenging datasets, demonstrate the competitiveness of our algorithms compared with other state-of-the-art methods.