arXiv research

This paper solves matrix blind joint block diagonalization with noise.

Diagonalizes metrics of 3D Lorentzian manifolds.

Study Ricci vector fields on 2D space with diagonal metrics.

Develops a novel stochastic algorithm for diagonal estimation of large matrices.

Octagon map accelerates diagonal changes algorithm.

Study finds symmetries in a special 3D space with a diagonal metric.

New diagonal knots found with non-torus structure.

We use mathematical induction to prove that the horizontal composition in the class of coherently diagonal complexes is indeed a binary operation. That is to say, the embedding of two coherently diagonal complexes in an alternating planar diagram produces a coherently diagonal complex.

Most known four-dimensional cohomogeneity-one Einstein metrics are diagonal in the basis defined by the left-invariant one-forms, though some essentially non-diagonal ones are known. We consider the problem of explicitly seeking non-diagonal Einstein metrics, and we find solutions which in some cases exhaust the possib…

Equal diagonal energies proven on Liouville surfaces.

Diagonal linear networks converge to lasso regularization path during training.

We show that a basis of a semisimple Lie algebra of compact type, for which any diagonal left-invariant metric has a diagonal Ricci tensor, is characterized by the Lie algebraic condition of being "nice". Namely, the bracket of any two basis elements is a multiple of another basis element. This extends the work of Laur…

New diagonal move simplifies knots and links efficiently.

We obtain the natural diagonal almost product and locally product structures on the total space of the cotangent bundle of a Riemannian manifold. We find the Riemannian almost product (locally product) and the (almost) para-Hermitian cotangent bundles of natural diagonal lift type. We prove the characterization theorem…

Study grid homology of diagonal knots, finding key terms related to prime factors and decompositions.

Study on stability of non-diagonal Einstein metrics on specific homogeneous spaces.

The author connects Poincaré embeddings to Reidemeister traces and diagonal maps.

Conditions for flat 3-manifolds with diagonal metrics are identified.

Constructs coordinates to diagonalize Toda flow on matrices with simple spectrum.

Adaptive gradient approaches that automatically adjust the learning rate on a per-feature basis have been very popular for training deep networks. This rich class of algorithms includes Adagrad, RMSprop, Adam, and recent extensions. All these algorithms have adopted diagonal matrix adaptation, due to the prohibitive co…

Paper proposes ABDR for convex subspace clustering with adaptive block diagonal representation.

This paper optimizes diagonal preconditioning to improve matrix condition numbers.

The approximate joint diagonalization of a set of matrices consists in finding a basis in which these matrices are as diagonal as possible. This problem naturally appears in several statistical learning tasks such as blind signal separation. We consider the diagonalization criterion studied in a seminal paper by Pham (…

We prove a number of convexity results for strata of the diagonal pants graph of a surface, in analogy with the extrinsic geometric properties of strata in the Weil-Petersson completion. As a consequence, we exhibit convex flat subgraphs of every possible rank inside the diagonal pants graph.

Classify projective subvarieties in Bogomolov-Guan manifolds using quasi-diagonals.

In this paper, we study deep diagonal circulant neural networks, that is deep neural networks in which weight matrices are the product of diagonal and circulant ones. Besides making a theoretical analysis of their expressivity, we introduced principled techniques for training these models: we devise an initialization s…

The main purpose of this note is to prove that any basis of a nilpotent Lie algebra for which all diagonal left-invariant metrics have diagonal Ricci tensor necessarily produce quite a simple set of structural constants; namely, the bracket of any pair of elements of the basis must be a multiple of some of them and onl…

The main goal of this paper is a calculation of the integral (co)homology of the group of symmetric automorphisms of a free product. We proceed by giving a geometric interpretation of symmetric automorphisms via a moduli space of certain diagrams, which we name cactus products. To describe this moduli space a theory of…

We prove a new off-diagonal asymptotic of the Bergman kernels associated to tensor powers of a positive line bundle on a compact Kähler manifold. We show that if the Kähler potential is real analytic, then the Bergman kernel accepts a complete asymptotic expansion in a neighborhood of the diagonal of shrinking size $k^…

Study of symmetries in a 2D space with specific metric properties.

Defines tensor products for A-infinity structures using diagonals.

New method improves deep learning model robustness and accuracy for long sequences.

New theory allows simultaneous block-diagonalization of commuting operator fields.

We give a quadratic lower bound on the dimension of the space of conjugacy classes of subgroups of SL(n,R) that are limits under conjugacy of the diagonal subgroup. We give the first explicit examples of abelian n-1 dimensional subgroups of SL(n,R) which are not such a limit, however all such abelian groups are limits …

Diagonal metrics solve Hermitian-Einstein equations for decomposed Higgs bundles.

In this paper, we propose a new Recurrent Neural Network (RNN) architecture. The novelty is simple: We use diagonal recurrent matrices instead of full. This results in better test likelihood and faster convergence compared to regular full RNNs in most of our experiments. We show the benefits of using diagonal recurrent…

The paper explores how sinks and diagonal patterns prevent attention oversmoothing.

We describe the geometric notion of distribution in synthetic terms, utilizing the notion of "first neighbourhood of the diagonal" from algebraic geometry. We characterize involutive distributions in combinatorial terms.

Study properties of 3D almost η-Ricci solitons with diagonal metrics.

We introduce the notion of Haantjes algebra: It consists of an assignment of a family of operator fields on a differentiable manifold, each of them with vanishing Haantjes torsion. They are also required to satisfy suitable compatibility conditions. Haantjes algebras naturally generalize several known interesting geome…

Diagonal transformations preserve independence structures in non-Gaussian distributions.

New insights into Hessian structure of neural networks reveal two forces.

In this article, we derive off-diagonal estimates of the Bergman kernel associated to tensor- products of the cotangent line bundle defined over a hyperbolic Riemann surface of finite volume.

New MCMC method learns sparse preconditioner for high-dimensional problems.

SGD on diagonal linear networks approximates to SDE in high dimensions.

Randomized block-diagonal preconditioning improves parallel learning convergence.

We propose a fast general projection-free metric learning framework, where the minimization objective $\min_{\textbf{M} \in \mathcal{S}} Q(\textbf{M})$ is a convex differentiable function of the metric matrix $\textbf{M}$, and $\textbf{M}$ resides in the set $\mathcal{S}$ of generalized graph Laplacian matrices for con…

We discuss a Lie algebraic and differential geometry construction of solutions to some multidimensional nonlinear integrable systems describing diagonal metrics on Riemannian manifolds, in particular those of zero and constant curvature. Here some special solutions to the Lamé and Bourlet type equations, determining by…