Related papers: Exact ReLU realization of tensor-product refinemen…

We study homogeneous refinement operators \((V\gamma)(t)=\sum_{j\in\mathbb Z}A_j\gamma(Mt-j)\), acting on compactly supported continuous piecewise linear curves \(\gamma:\mathbb R\to\mathbb R^p\), where \(M\ge2\) and only finitely many…

Ardakov-Wadsley defined the sheaf D-cap of $p$-adic analytic differential operators on a smooth rigid analytic variety $X$ by restricting to the case where $X$ is affinoid and the tangent sheaf admits a smooth Lie lattice. We generalize…

Formal verification of transformers has become increasingly important due to their widespread deployment in safety-critical applications. Compared to classic neural networks, the inferences of transformers involve highly complex…

Using the notions and tools from realization in the sense of systems theory, we establish an explicit and new realization formula for families of infinite products of rational matrix-functions of a single complex variable. Our realizations…

Accurate finite element analysis of refined shell theories is crucial but often hindered by membrane and shear locking effects. While various element-based locking-free techniques exist, this work addresses the problem at the theoretical…

A recent line of work shows that a deep neural network with ReLU nonlinearities arises from a finite sequence of cascaded sparse coding models, the outputs of which, except for the last element in the cascade, are sparse and unobservable.…

Let K=Fq be a finite field. We introduce a family of projective Reed-Muller-type codes called projective Segre codes. Using commutative algebra and linear algebra methods, we study their basic parameters and show that they are direct…

The linear layer is one of the most pervasive modules in deep learning representations. However, it requires $O(N^2)$ parameters and $O(N^2)$ operations. These costs can be prohibitive in mobile applications or prevent scaling in many…

Deep neural networks, particularly those employing Rectified Linear Units (ReLU), are often perceived as complex, high-dimensional, non-linear systems. This complexity poses a significant challenge to understanding their internal learning…

In order to have a multiresolution analysis, the scaling function must be refinable. That is, it must be the linear combination of 2-dilation, $\mathbb{Z}$-translates of itself. Refinable functions used in connection with wavelets are…

This paper conducts a rigorous analysis for provable estimation of multidimensional arrays, in particular third-order tensors, from a random subset of its corrupted entries. Our study rests heavily on a recently proposed tensor algebraic…

Tensor completion recovers a multi-dimensional array from a limited number of measurements. Using the recently proposed tensor ring (TR) decomposition, in this paper we show that a d-order tensor of dimensional size n and TR rank r can be…

This paper is concerned with the low Tucker-rank tensor completion problem, which is about reconstructing a tensor $ T \in\mathbb{R}^{n\times n \times n}$ of low multilinear rank from partially observed entries. Riemannian optimization…

We study the problem of inverting a deep generative model with ReLU activations. Inversion corresponds to finding a latent code vector that explains observed measurements as much as possible. In most prior works this is performed by…

Bounded model checking is among the most efficient techniques for the automatic verification of concurrent programs. However, encoding all possible interleavings often requires a huge and complex formula, which significantly limits the…

Tensor completion is the problem of estimating the missing values of high-order data from partially observed entries. Data corruption due to prevailing outliers poses major challenges to traditional tensor completion algorithms, which…

Many nonlinear optimal control and optimization problems involve constraints that combine continuous dynamics with discrete logic conditions. Standard approaches typically rely on mixed-integer programming, which introduces scalability…

Robustness verification is a promising technique for rigorously proving Recurrent Neural Networks (RNNs) robustly. A key challenge is to over-approximate the nonlinear activation functions with linear constraints, which can transform the…

In the framework of multidimensional Compressed Sensing (CS), we introduce an analytical reconstruction formula that allows one to recover an $N$th-order $(I_1\times I_2\times \cdots \times I_N)$ data tensor $\underline{\mathbf{X}}$ from a…

We study properties of a C_2-cofinite vertex operator algebra of CFT type. If it is also rational and V'\cong V, then the rigidity of the tensor category of modules has been proved by Huang. When we treat an irrational C_2-cofinite VOA, the…