Related papers: Polarization of neural codes
Polar coding, introduced 2008 by Arikan, is the first (very) efficiently encodable and decodable coding scheme whose information transmission rate provably achieves the Shannon bound for classical discrete memoryless channels in the…
Understanding the nature of representation in neural networks is a goal shared by neuroscience and machine learning. It is therefore exciting that both fields converge not only on shared questions but also on similar approaches. A pressing…
The study of the brain's representations of uncertainty is a central topic in neuroscience. Unlike most quantities of which the neural representation is studied, uncertainty is a property of an observer's beliefs about the world, which…
This paper presents a definition for local linearizations of rational matrices and studies their properties. This definition allows us to introduce matrix pencils associated to a rational matrix that preserve its structure of zeros and…
We seek to improve deep neural networks by generalizing the pooling operations that play a central role in current architectures. We pursue a careful exploration of approaches to allow pooling to learn and to adapt to complex and variable…
This dissertation explores the roles of polarities and focussing in various aspects of Computational Logic. These concepts play a key role in the the interpretation of proofs as programs, a.k.a. the Curry-Howard correspondence, in the…
The human brain cortical layer has a convoluted morphology that is unique to each individual. Characterization of the cortical morphology is necessary in longitudinal studies of structural brain change, as well as in discriminating…
An old idea in optimization theory says that since the gradient is a dual vector it may not be subtracted from the weights without first being mapped to the primal space where the weights reside. We take this idea seriously in this paper…
Arikan's Polar codes attracted much attention as the first efficiently decodable and capacity achieving codes. Furthermore, Polar codes exhibit an exponentially decreasing block error probability with an asymptotic error exponent upper…
In the realm of computer vision, Neural Fields have gained prominence as a contemporary tool harnessing neural networks for signal representation. Despite the remarkable progress in adapting these networks to solve a variety of problems,…
Let $D\subseteq B$ be an extension of integral domains and $E$ a subset of the quotient field of $D$. We introduce the ring of \textit{$D$-valued $B$-rational functions on $E$}, denoted by $Int^R_B(E,D)$, which naturally extends the…
An ideal of a local polynomial ring can be described by calculating a standard basis with respect to a local monomial ordering. However standard basis algorithms are not numerically stable. Instead we can describe the ideal numerically by…
We generalize the theory of logarithmic derivations through a self-contained study of modules here dubbed tangential idealizers. We establish reflexiveness criteria for such modules, provided the ring is a factorial domain. As a main…
We introduce the factor complex of a neural code, and show how intervals and maximal codewords are captured by the combinatorics of factor complexes. We use these results to obtain algebraic and combinatorial characterizations of…
This paper investigates the impact of puncturing on finite-length polar codes in which a puncturing pattern $\pv^{N}=(p_0,...,p_N)$ is applied to a length-$N$ polar code.. We first introduce two virtual channels to stochastically model the…
The concept of hypergroup is generalization of group, first was introduced by Marty [9]. This theory had applications to several domains. Marty had applied them to groups, algebraic functions and rational functions. M. Krasner has studied…
Neural Language Models of Code, or Neural Code Models (NCMs), are rapidly progressing from research prototypes to commercial developer tools. As such, understanding the capabilities and limitations of such models is becoming critical.…
The main purpose of this paper is to develop new algorithms for computing invariant rings in a general setting. This includes invariants of nonreductive groups but also of groups acting on algebras over certain rings. In particular, we…
This paper proposes to study neural networks through neuronal correlation, a statistical measure of correlated neuronal activity on the penultimate layer. We show that neuronal correlation can be efficiently estimated via weight matrix, can…
This work revisits operator learning from a spectral perspective by introducing Polar Linear Algebra, a structured framework based on polar geometry that combines a linear radial component with a periodic angular component. Starting from…