Related papers: Asymptotic rank bounds: a numerical census

By means of two simple convexity arguments we are able to develop a general method for proving consistency and asymptotic normality of estimators that are defined by minimisation of convex criterion functions. This method is then applied to…

In this paper we prove the asymptotic efficiency of the model selection procedure proposed by the authors in the first part. To this end we introduce the robust risk as the least upper bound of the quadratical risk over a broad class of…

Motivated by problems in algebraic complexity theory (e.g., matrix multiplication) and extremal combinatorics (e.g., the cap set problem and the sunflower problem), we introduce the geometric rank as a new tool in the study of tensors and…

In this article we define and study a notion of asymptotic rank for metric spaces and show in our main theorem that for a large class of spaces, the asymptotic rank is characterized by the growth of the higher filling functions. For a…

We compute the asymptotic induced matching number of the $k$-partite $k$-uniform hypergraphs whose edges are the $k$-bit strings of Hamming weight $k/2$, for any large enough even number $k$. Our lower bound relies on the higher-order…

We consider tensor factorizations based on sparse measurements of the components of relatively high rank tensors. The measurements are designed in a way that the underlying graph of interactions is a random graph. The setup will be useful…

Matrix completion algorithms recover a low rank matrix from a small fraction of the entries, each entry contaminated with additive errors. In practice, the singular vectors and singular values of the low rank matrix play a pivotal role for…

Asymptotic expansion is constructed and justified for the solution to a nonuniform Neumann boundary-value problem for the Poisson equation with the right-hand side that depends both on longitudinal and transversal variables in a thin…

For tensors of fixed order, we establish three types of upper bounds for the geometric rank in terms of the subrank. Firstly, we prove that, under a mild condition on the characteristic of the base field, the geometric rank of a tensor is…

We establish asymptotic bounds for the number of partitions of $[n]$ avoiding a given partition in Klazar's sense, obtaining the correct answer to within an exponential for the block case. This technique also enables us to establish a…

We herein establish an asymptotic representation theorem for locally asymptotically normal quantum statistical models. This theorem enables us to study the asymptotic efficiency of quantum estimators such as quantum regular estimators and…

We present a family of flattening methods of tensors which we call Kronecker-Koszul flattenings, generalizing the famous Koszul flattenings and further equations of secant varieties studied among others by Landsberg, Manivel, Ottaviani and…

We study quantum dichotomies and the resource theory of asymmetric distinguishability using a generalization of Strassen's theorem on preordered semirings. We find that an asymptotic variant of relative submajorization, defined on…

In this paper, we consider the statistical inference for several low-rank tensor models. Specifically, in the Tucker low-rank tensor PCA or regression model, provided with any estimates achieving some attainable error rate, we develop the…

Higher-order tensor datasets arise commonly in recommendation systems, neuroimaging, and social networks. Here we develop probable methods for estimating a possibly high rank signal tensor from noisy observations. We consider a generative…

Rank estimation is a classical model order selection problem that arises in a variety of important statistical signal and array processing systems, yet is addressed relatively infrequently in the extant literature. Here we present sample…

Symbolic regression is a type of discrete optimization problem that involves searching expressions that fit given data points. In many cases, other mathematical constraints about the unknown expression not only provide more information…

We address the problem of the additivity of the tensor rank. That is for two independent tensors we study if the rank of their direct sum is equal to the sum of their individual ranks. A positive answer to this problem was previously known…

In this paper, we investigate the sample complexity of recovering tensors with low symmetric rank from symmetric rank-one measurements. This setting is particularly motivated by the study of higher-order interactions and the analysis of…

In this work, we improve upon the guarantees for sparse random embeddings, as they were recently provided and analyzed by Freksen at al. (NIPS'18) and Jagadeesan (NIPS'19). Specifically, we show that (a) our bounds are explicit as opposed…