Related papers: Reconstruction of modular data from $SL_2(\mathbb{…

We develop a framework to analyse invariant decompositions of elements of tensor product spaces. Namely, we define an invariant decomposition with indices arranged on a simplicial complex, and which is explicitly invariant under a group…

Data augmentation has been proven to be an effective technique for developing machine learning models that are robust to known classes of distributional shifts (e.g., rotations of images), and alignment regularization is a technique often…

Tensors or multiarray data are generalizations of matrices. Tensor clustering has become a very important research topic due to the intrinsically rich structures in real-world multiarray datasets. Subspace clustering based on vectorizing…

In this paper we propose novel methods for compression and recovery of multilinear data under limited sampling. We exploit the recently proposed tensor- Singular Value Decomposition (t-SVD)[1], which is a group theoretic framework for…

It is a well-known result of Etingof, Nikshych and Ostrik that there are finitely many inequivalent integral modular categories of any fixed rank $n$. This follows from a double-exponential bound on the maximal denominator in an Egyptian…

This paper classifies all modular data of integral modular fusion categories up to rank 13. Furthermore, it also classifies all integral half-Frobenius fusion rings up to rank 12. We find that each perfect integral modular fusion category…

Modular tensor categories are generalizations of the representation categories of quantum groups at roots of unity axiomatizing the properties necessary to produce 3-dimensional TQFTs. Although other constructions have since been found,…

Machine learning and data mining algorithms are becoming increasingly important in analyzing large volume, multi-relational and multi--modal datasets, which are often conveniently represented as multiway arrays or tensors. It is therefore…

A new submodule clustering method via sparse and low-rank representation for multi-way data is proposed in this paper. Instead of reshaping multi-way data into vectors, this method maintains their natural orders to preserve data intrinsic…

Determining the matrix multiplication exponent $\omega$ is one of the greatest open problems in theoretical computer science. We show that it is impossible to prove $\omega = 2$ by starting with structure tensors of modules of fixed degree…

We study finite-rank left-translation invariant algebraic $D$-modules on complex affine algebraic groups. Using the standard description of these objects as left-invariant flat algebraic connections on the trivial vector bundle, modulo…

This paper introduces Modular Linear Tokenization (MLT), a reversible and deterministic technique for encoding high-cardinality categorical identifiers into compact numerical vectors. Unlike traditional hashing or one-hot encodings, MLT…

We propose a novel memory-modular learner for image classification that separates knowledge memorization from reasoning. Our model enables effective generalization to new classes by simply replacing the memory contents, without the need for…

We classify semisimple module categories over the tensor category of representations of quantum SL(2) extending previous results to the roots of unity and positive characteristic cases.

The factorization of three-dimensional data continues to gain attention due to its relevance in representing and compressing large-scale datasets. The linear-map-based tensor-tensor multiplication is a matrix-mimetic operation that extends…

It has been hypothesized that some form of "modular" structure in artificial neural networks should be useful for learning, compositionality, and generalization. However, defining and quantifying modularity remains an open problem. We cast…

The growing prevalence of tensor data, or multiway arrays, in science and engineering applications motivates the need for tensor decompositions that are robust against outliers. In this paper, we present a robust Tucker decomposition…

Owing to the cross-pollination between causal discovery and deep learning, non-statistical data (e.g., images, text, etc.) encounters significant conflicts in terms of properties and methods with traditional causal data. To unify these data…

The main goal of this paper is to classify $\ast$-module categories for the $SO(3)_{2m}$ modular tensor category. This is done by classifying $SO(3)_{2m}$ nimrep graphs and cell systems, and in the process we also classify the $SO(3)$…

We propose a modular framework for multi-relational learning via tensor decomposition. In our learning setting, the training data contains multiple types of relationships among a set of objects, which we represent by a sparse three-mode…