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We present a new distributed representation in deep neural nets wherein the information is represented in native form as a matrix. This differs from current neural architectures that rely on vector representations. We consider matrices as…
This article examines matrices whose entries are determined by recursive relations of the form $A_{i, j} = x A_{i, j-1} + y A_{i-1, j-1} + z A_{i-1, j}$, where $x, y, z$ are constants, and the initial conditions are defined along the first…
In this note, after recalling a proof of the Macdonald identities for untwisted affine root systems, we derive the Macdonald identities for twisted affine root systems.
This paper focuses on the study of recognizing discontiguous entities. Motivated by a previous work, we propose to use a novel hypergraph representation to jointly encode discontiguous entities of unbounded length, which can overlap with…
A formal computation proving a new operator identity from known ones is, in principle, restricted by domains and codomains of linear operators involved, since not any two operators can be added or composed. Algebraically, identities can be…
The objective of this work is to learn a compact embedding of a set of descriptors that is suitable for efficient retrieval and ranking, whilst maintaining discriminability of the individual descriptors. We focus on a specific example of…
The use of high-dimensional features has become a normal practice in many computer vision applications. The large dimension of these features is a limiting factor upon the number of data points which may be effectively stored and processed,…
Case-based explanations are an intuitive method to gain insight into the decision-making process of deep learning models in clinical contexts. However, medical images cannot be shared as explanations due to privacy concerns. To address this…
This paper proposes a novel matrix rank-one decomposition for quaternion Hermitian matrices, which admits a stronger property than the previous results in (sturm2003cones,huang2007complex,ai2011new). The enhanced property can be used to…
Face anonymization aims to conceal the visual identity of a face to safeguard the individual's privacy. Traditional methods like blurring and pixelation can largely remove identifying features, but these techniques significantly degrade…
We find new examples of the $P=W$ identity of de Cataldo-Hausel-Migliorini by studying cluster varieties. We prove that the weight filtration of 2D cluster varieties correspond to the perverse filtration of elliptic fibrations which are…
For a one-parameter degeneration of reduced compact complex analytic spaces of dimension $n$, we prove the invariance of the frontier Hodge numbers $h^{p,q}$ (that is, with $pq(n{-}p)(n{-}q)=0$) for the intersection cohomology of the fibers…
We consider square matrices over $\mathbb{C}$ satisfying an identity relating their eigenvalues and the corresponding eigenvectors re-proved and discussed by Denton, Parker, Tao and Zhang, called the eigenvector-eigenvalue identity. We…
A superposition of a matrix ensemble refers to the ensemble constructed from two independent copies of the original, while a decimation refers to the formation of a new ensemble by observing only every second eigenvalue. In the cases of the…
We introduce dilogarithm identities through a beta integral-based technique that we apply to provide analytic proofs of previously conjectured dilogarithm relations, solving open problems given by both Bytsko and Campbell, and that we…
We propose a framework to analyze how multivariate representations disentangle ground-truth generative factors. A quantitative analysis of disentanglement has been based on metrics designed to compare how one variable explains each…
Let X be a singular affine normal variety with coordinate ring R and assume that there is an R-order admitting a stability structure such that the scheme of relevant semistable representations is smooth, then we construct a partial…
In a recent paper, Caracciolo, Sokal and Sportiello presented, inter alia, an algebraic/combinatorial proof for Cayley's identity. The purpose of the present paper is to give a "purely combinatorial" proof for this identity; i.e., a proof…
Recently, we have seen an increase in the global facial recognition market size. Despite significant advances in face recognition technology with the adoption of convolutional neural networks, there are still open challenges, such as when…
We study non-associative twisted group algebras over $(\Z_2)^n$ with cubic twisting functions. We construct a series of algebras that extend the classical algebra of octonions in the same way as the Clifford algebras extend the algebra of…