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Choosing an appropriate regularization term is necessary to obtain a meaningful solution to an ill-posed linear inverse problem contaminated with measurement errors or noise. The $\ell_p$ norm covers a wide range of choices for the…

In representation learning (RL), how to make the learned representations easy to interpret and less overfitted to training data are two important but challenging issues. To address these problems, we study a new type of regulariza- tion…

Recently, we introduced Relative Resolution as a hybrid formalism for fluid mixtures [1]. The essence of this approach is that it switches molecular resolution in terms or relative separation: While nearest neighbors are characterized by a…

Inductive reasoning, a cornerstone of human cognition, enables generalization from limited data but hasn't yet been fully achieved by large language models (LLMs). While modern LLMs excel at reasoning tasks, their ability to maintain stable…

The problem of numerical differentiation can be thought of as an inverse problem by considering it as solving a Volterra equation. It is well known that such inverse integral problems are ill-posed and one requires regularization methods to…

Logical bilateralism challenges traditional concepts of logic by treating assertion and denial as independent yet opposed acts. While initially devised to justify classical logic, its constructive variants show that both acts admit…

REgularization by Denoising (RED) is an attractive framework for solving inverse problems by incorporating state-of-the-art denoising algorithms as the priors. A drawback of this approach is the high computational complexity of denoisers,…

The theorem of factorisation forests shows the existence of nested factorisations -- a la Ramsey -- for finite words. This theorem has important applications in semigroup theory, and beyond. The purpose of this paper is to illustrate the…

We present a counter-example to the statement that there is a linear reduction in both directionsbetween the largest common subtree and the smallest common supertrees. Moreover, we show that in general, even having of one of them, it is…

In this paper, we investigate the principle that `good explanations are hard to vary' in the context of deep learning. We show that averaging gradients across examples -- akin to a logical OR of patterns -- can favor memorization and…

In arXiv:2204.03190, we proposed a universal method to reduce one-loop integrals with both tensor structure and higher-power propagators. But the method is quite redundant as it does not utilize the results of lower rank cases when…

In this work, a set reconciliation setting is considered in which two parties have similar sets that they would like to reconcile. In particular, we focus on a divide-and-conquer strategy known as partitioned set reconciliation (PSR), in…

In our previous papers we sketched proofs of the equality NP = coNP = PSPACE. These results have been obtained by proof theoretic tree-to-dag compressing techniques adapted to Prawitz's Natural Deduction (ND) for implicational minimal logic…

Deep learning has been successfully applied to various tasks, but its underlying mechanism remains unclear. Neural networks associate similar inputs in the visible layer to the same state of hidden variables in deep layers. The fraction of…

In this paper, we establish robustness to noise perturbations of polyhedral regularization of linear inverse problems. We provide a sufficient condition that ensures that the polyhedral face associated to the true vector is equal to that of…

We initiate a program of parameterized proof complexity that aims to provide evidence that FPT is different from W[1]. A similar program already exists for the classes W[2] and W[SAT]. We contrast these programs and prove upper and lower…

Based on a reduction processing, we rewrite a hypergeometric term as the sum of the difference of a hypergeometric term and a reduced hypergeometric term (the reduced part, in short). We show that when the initial hypergeometric term has a…

Given an undirected graph representing similarities between a set of items and an additive measure evaluating the items, we treat the position of a special subset of items in an ordinal ranking through a collection of combinatorial…

Higher-order representations of objects such as programs, proofs, formulas and types have become important to many symbolic computation tasks. Systems that support such representations usually depend on the implementation of an intensional…

We consider extensions of the language of Peano arithmetic by transfinitely iterated truth definitions satisfying uniform Tarskian biconditionals. Without further axioms, such theories are known to be conservative extensions of the original…