Related papers: Probabilistic Rank and Matrix Rigidity
Low-rank tensor approximations have shown great potential for uncertainty quantification in high dimensions, for example, to build surrogate models that can be used to speed up large-scale inference problems (Eigel et al., Inverse Problems…
This paper considers the problem of matrix-variate logistic regression. It derives the fundamental error threshold on estimating low-rank coefficient matrices in the logistic regression problem by obtaining a lower bound on the minimax…
Design matrices are sparse matrices in which the supports of different columns intersect in a few positions. Such matrices come up naturally when studying problems involving point sets with many collinear triples. In this work we consider…
We prove new upper and lower bounds on $\epsilon$-approximate sign-rank, a relaxation of sign-rank introduced by Chornomaz, Moran, and Waknine (STOC 2025). We show that every $m \times n$ sign matrix with approximate sign-rank $d$ contains…
In this paper, we investigate computational power of threshold circuits and other theoretical models of neural networks in terms of the following four complexity measures: size (the number of gates), depth, weight and energy. Here the…
The communication class $\mathbf{UPP}^{\text{cc}}$ is a communication analog of the Turing Machine complexity class $\mathbf{PP}$. It is characterized by a matrix-analytic complexity measure called sign-rank (also called dimension…
In 2024, Kliuchnikov and Sch\"onnenbeck showed a connection between the Barnes Wall lattices, stabilizer states and Clifford operations. In this work, we study their results and relate them to the problem of lower bounding stabilizer ranks.…
With the growth of model and data sizes, a broad effort has been made to design pruning techniques that reduce the resource demand of deep learning pipelines, while retaining model performance. In order to reduce both inference and training…
We introduce the polynomial coefficient matrix and identify maximum rank of this matrix under variable substitution as a complexity measure for multivariate polynomials. We use our techniques to prove super-polynomial lower bounds against…
A well studied problem in algebraic complexity theory is the determination of the complexity of problems relying on evaluations of bilinear maps. One measure of the complexity of a bilinear map (or 3-tensor) is the optimal number of…
The development of randomized algorithms for numerical linear algebra, e.g. for computing approximate QR and SVD factorizations, has recently become an intense area of research. This paper studies one of the most frequently discussed…
The log-rank conjecture is one of the fundamental open problems in communication complexity. It speculates that the deterministic communication complexity of any two-party function is equal to the log of the rank of its associated matrix,…
Variational minimization of the ground-state energy as a function of the two-electron reduced density matrix (2-RDM), constrained by necessary $N$-representability conditions, provides a polynomial-scaling approach to studying strongly…
The notion of `stable rank' of a matrix is central to the analysis of randomized matrix algorithms, covariance estimation, deep neural networks, and recommender systems. We compare the properties of the stable rank and intrinsic dimension…
Low-rank approximation of a matrix by means of random sampling has been consistently efficient in its empirical studies by many scientists who applied it with various sparse and structured multipliers, but adequate formal support for this…
Finding whether a linear-constraint loop has a linear ranking function is an important key to understanding the loop behavior, proving its termination and establishing iteration bounds. If no preconditions are provided, the decision problem…
In this paper we apply methods originated in Complexity theory to some problems of Approximation. We notice that the construction of Alman and Williams that disproves the rigidity of Walsh-Hadamard matrices, provides good…
Let A be an n*n random matrix with mean zero and independent inhomogeneous non-constant subgaussian entries. We get that for any k<c\sqrt{n}, the probability of the matrix has a lower rank than n-k that is sub-exponential. Furthermore, we…
By a tensor we mean an element of a tensor product of vector spaces over a field. Up to a choice of bases in factors of tensor products, every tensor may be coordinatized, that is, represented as an array consisting of numbers. This note is…
Recently, fundamental conditions on the sampling patterns have been obtained for finite completability of low-rank matrices or tensors given the corresponding ranks. In this paper, we consider the scenario where the rank is not given and we…