We formulate the Riemannian calculus of the probability set embedded with L^2-Wasserstein metric. This is an initial work of transport information geometry. Our investigation starts with the probability simplex (probability manifold) supported on vertices of a finite graph. The main idea is to embed the probability manifold as a submanifold of the positive measure space with a nonlinear metric tensor. Here the nonlinearity comes from the linear weighted Laplacian operator. By this viewpoint, we establish torsion-free Christoffel symbols, Levi-Civita connections, curvature tensors, and volume forms in the probability manifold by Euclidean coordinates. As a consequence, the Jacobi equation, Laplace-Beltrami and Hessian operators on the probability manifold are derived. These geometric computations are also provided in the infinite-dimensional density space (density manifold) supported on a finite-dimensional manifold. In particular, an identity is given connecting the Baker-{É}mery Γ2 operator (carr{é} du champ it{é}r{é}) by connecting Fisher-Rao information metric and optimal transport metric. Several examples are demonstrated.

The moment-entropy inequality shows that a contin- uous random variable with given second moment and maximal Shannon entropy must be Gaussian. Stam’s inequality shows that a continuous random variable with given Fisher information and minimal Shannon entropy must also be Gaussian. The Crame ́r- Rao inequality is a direct consequence of these two inequalities. In this paper the inequalities above are extended to Renyi entropy, p-th moment, and generalized Fisher information. Gen- eralized Gaussian random densities are introduced and shown to be the extremal densities for the new inequalities. An extension of the Crame ́r–Rao inequality is derived as a consequence of these moment and Fisher information inequalities.

We define the notion of a Ricci curvature lower bound for parametrized statistical models. Following the seminal ideas of Lott–Sturm–Villani, we define this notion based on the geodesic convexity of the Kullback–Leibler divergence in a Wasserstein statistical manifold, that is, a manifold of probability distributions endowed with a Wasserstein metric tensor structure. Within these definitions, which are based on Fisher information matrix and Wasserstein Christoffel symbols, the Ricci curvature is related to both, information geometry and Wasserstein geometry. These definitions allow us to formulate bounds on the convergence rate of Wasserstein gradient flows and information functional inequalities in parameter space. We discuss examples of Ricci curvature lower bounds and convergence rates in exponential family models.

In this paper we integrate semi-local patches and the weighted graph Laplacian into the framework of the low dimensional manifold model. This approach is much faster than the original LDMM algorithm. The number of iterations is typically reduced from 100 to 10 and the equations in each step are much easier to solve. This new approach is tested in image inpainting and denoising and the results are better than the original LDMM and competitive with state-of-the-art methods.

We present the application of a low dimensional manifold model (LDMM) on hyperspectral image (HSI) reconstruction. An important property of hyperspectral images is that the patch manifold, which is sampled by the three-dimensional blocks in the data cube, is generally of a low dimensional nature. This is a generalization of low-rank models in that hyperspectral images with nonlinear mixing terms can also fit in this framework. The point integral method (PIM) is used to solve a Laplace-Beltrami equation over a point cloud sampling the patch manifold in LDMM. Both numerical simulations and theoretical analysis show that the sample points constraint is correctly enforced by PIM. The framework is demonstrated by experiments on the reconstruction of both linear and nonlinear mixed hyperspectral images with a significant number of missing voxels and several entirely missing spectral bands.

In this paper, we prove that the degree of regularity of square systems, a subfamily of the HFE systems, over a prime finite field of odd characteristic q is exactly q and, therefore, prove that inverting square systems algebraically using Gröbner basis algorithm is exponential, when q = \Omega(n), where n is the number of variables of the system.

A paper was published (Harsha and Subrahamanian Moosath, 2014) in which the authors claimed to have discovered an extension to Amari's \(\alpha\)-geometry through a general monotone embedding function. It will be pointed out here that this so-called \((F, G)\)-geometry (which includes \(F\)-geometry as a special case) is identical to Zhang's (2004) extension to the \(\alpha\)-geometry, where the name of the pair of monotone embedding functions \(\rho\) and \(\tau\) were used instead of \(F\) and \(H\) used in Harsha and Subrahamanian Moosath (2014). Their weighting function \(G\) for the Riemannian metric appears cosmetically due to a rewrite of the score function in log-representation as opposed to \((\rho, \tau)\)-representation in Zhang (2004). It is further shown here that the resulting metric and \(\alpha\)-connections obtained by Zhang (2004) through arbitrary monotone embeddings is a unique extension of the \(\alpha\)-geometric structure. As a special case, Naudts' (2004) \(\phi\)-logarithm embedding (using the so-called \(\log_\phi\) function) is recovered with the identification \(\rho=\phi, \, \tau=\log_\phi\), with \(\phi\)-exponential \(\exp_\phi\) given by the associated convex function linking the two representations.

A weighted and convex regularized nuclear norm model is introduced to construct a rank constrained linear transform on feature vectors of deep neural networks. The feature vectors of each class are modeled by a subspace, and the linear transform aims to enlarge the pairwise angles of the subspaces. The weight and convex regularization resolve the rank degeneracy of the linear transform. The model is computed by a dierence of convex function algorithm whose descent and convergence properties are analyzed. Numerical experiments are carried out in convolutional neural networks on CAFFE platform for 10 class handwritten digit images (MNIST) and small object color images (CIFAR-10) in the public domain. The transformed feature vectors improve the accuracy of the network in the regime of low dimensional features subsequent to principal component analysis. The feature transform is independent of the network structure, and can be applied without retraining the feature extraction layers of the network.

In this paper, we propose a novel low dimensional manifold model (LDMM) and apply it to some image processing problems. LDMM is based on the fact that the patch manifolds of many natural images have low dimensional structure. Based on this fact, the dimension of the patch manifold is used as a regularization to recover the image. The key step in LDMM is to solve a Laplace-Beltrami equation over a point cloud which is solved by the point integral method. The point integral method enforces the sample point constraints correctly and gives better results than the standard graph Laplacian. Numerical simulations in image denoising, inpainting and super-resolution problems show that LDMM is a powerful method in image processing.

Divergence functions are the non-symmetric “distance” on the manifold, Μθ, of parametric probability density functions over a measure space, ( Χ,μ). Classical information geometry prescribes, on Μθ: (i) a Riemannian metric given by the Fisher information; (ii) a pair of dual connections (giving rise to the family of α-connections) that preserve the metric under parallel transport by their joint actions; and (iii) a family of divergence functions ( α-divergence) defined on Μθ x？ Μθ, which induce the metric and the dual connections. Here, we construct an extension of this differential geometric structure from Μθ (that of parametric probability density functions) to the manifold, Μ, of non-parametric functions on X, removing the positivity and normalization constraints. The generalized Fisher information and α-connections on M are induced by an α-parameterized family of divergence functions, reflecting the fundamental convex inequality associated with any smooth and strictly convex function. The infinite-dimensional manifold, M, has zero curvature for all these α-connections; hence, the generally non-zero curvature of M can be interpreted as arising from an embedding of Μθ into Μ. Furthermore, when a parametric model (after a monotonic scaling) forms an affine submanifold, its natural and expectation parameters form biorthogonal coordinates, and such a submanifold is dually flat for α？= ± 1, generalizing the results of Amari’s α-embedding. The present analysis illuminates two different types of duality in information geometry, one concerning the referential status of a point (measurable function) expressed in the divergence function (“referential duality”) and the other concerning its representation under an arbitrary monotone scaling (“representational duality”).