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In this work, we focus on the task of learning and representing dense correspondences in deformable object categories. While this problem has been considered before, solutions so far have been rather ad-hoc for specific object types (i.e.,…
We obtain sufficient conditions ensuring the topological equivalence of two perturbed difference linear systems whose linear part has a property of generalized exponential dichotomy. When the exponential dichotomy is verified, we obtain a…
We study equivariant modules over $GL(V)$ over the polynomial ring $R = Sym V$. We introduce for every partition $\lambda$ the elementary equivariant module $M_{\lambda}$. Then we prove that any finitely generated equivariant module admits…
A differential geometric approach to singular perturbation theory is presented. It is shown that singular perturbation problems such as multiple-scale and boundary layer problems can be treated more easily on a differential geometric basis.…
Can we effectively learn a nonlinear representation in time comparable to linear learning? We describe a new algorithm that explicitly and adaptively expands higher-order interaction features over base linear representations. The algorithm…
These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new…
Special kinds of rank 2 vector bundles with (possibly irregular) connections on P^1 are considered. We construct an equivalence between the derived category of quasi-coherent sheaves on the moduli stack of such bundles and the derived…
A relationship between curved differential algebras and corings is established and explored. In particular it is shown that the category of semi-free curved differential graded algebras is equivalent to the category of corings with…
Generative models have advantageous characteristics for classification tasks such as the availability of unsupervised data and calibrated confidence, whereas discriminative models have advantages in terms of the simplicity of their model…
In this paper we consider graphs whose edges are associated with a degree of {\em importance}, which may depend on the type of connections they represent or on how recently they appeared in the scene, in a streaming setting. The goal is to…
Amenable category is a variant of the Lusternik-Schnirelman category, based on covers by amenable open subsets. We study the monotonicity problem for degree-one maps and amenable category and the relation between amenable category and…
Equivariance to symmetries has proven to be a powerful inductive bias in deep learning research. Recent works on mesh processing have concentrated on various kinds of natural symmetries, including translations, rotations, scaling, node…
In this paper a theory of reflective X-ray multilayer structures with a graded (slowly varying) period based on the coupled waves method and quasi-classical asymptotic expansions is reported. A number of exact solutions of the coupled wave…
Although Recurrent Neural Network (RNN) has been a powerful tool for modeling sequential data, its performance is inadequate when processing sequences with multiple patterns. In this paper, we address this challenge by introducing a novel…
We show that the mapping class group of an orientable finite type surface has uniformly exponential growth, as well as various closely related groups. This provides further evidence that mapping class groups may be linear.
We classify a class of infinite-dimensional simple graded pre-Lie algebras on the graded vector space underlying the algebra of Laurent polynomials, with a specific form for the product.
Several recent works demonstrate that transformers can implement algorithms like gradient descent. By a careful construction of weights, these works show that multiple layers of transformers are expressive enough to simulate iterations of…
We classify the solutions to an overdetermined elliptic problem in the plane in the finite connectivity case. This is achieved by establishing a one-to-one correspondence between the solutions to this problem and a certain type of minimal…
Scalable sequence models, such as Transformer variants and structured state-space models, often trade expressivity power for sequence-level parallelism, which enables efficient training. Here we examine the bounds on error and how error…
State-of-the-art training algorithms for deep learning models are based on stochastic gradient descent (SGD). Recently, many variations have been explored: perturbing parameters for better accuracy (such as in Extragradient), limiting SGD…