Related papers: Neural Codes and the Factor Complex
Matrix factorization is an important mathematical problem encountered in the context of dictionary learning, recommendation systems and machine learning. We introduce a new `decimation' scheme that maps it to neural network models of…
A covering code is a set of codewords with the property that the union of balls, suitably defined, around these codewords covers an entire space. Generally, the goal is to find the covering code with the minimum size codebook. While most…
Framed combinatorial topology is a novel theory describing combinatorial phenomena arising at the intersection of stratified topology, singularity theory, and higher algebra. The theory synthesizes elements of classical combinatorial…
The "neural code" is the way the brain characterizes, stores, and processes information. Unraveling the neural code is a key goal of mathematical neuroscience. Topology, coding theory, and, recently, commutative algebra are some the…
This paper concerns non-overlapping codes, block codes motivated by synchronisation and DNA-based storage applications. Most existing constructions of these codes do not account for the restrictions posed by the physical properties of…
A Code loop on a binary linear code that is doubly even with a factor set is shown to be a central loop, conjugacy closed loop, Burn loop and extra loop. General forms of the identities that define the factor set of a code are deduced.
Binary codes are constructed from incidence matrices of hypergraphs. A combinatroial description is given for the minimum distances of such codes via a combinatorial tool called ``eonv". This combinatorial approach provides a faster…
We introduce a new neural architecture and an unsupervised algorithm for learning invariant representations from temporal sequence of images. The system uses two groups of complex cells whose outputs are combined multiplicatively: one that…
We introduce "fractalization", a procedure by which spin models are extended to higher-dimensional "fractal" spin models. This allows us to interpret type-II fracton phases, fractal symmetry-protected topological phases, and more, in terms…
Hypergraphs are structures that can be decomposed or described; in other words they are recursively countable. Here, we get exact and asymptotic enumeration results on hypergraphs by means of exponential generating functions. The number of…
Combinatorial optimization can be described as the problem of finding a feasible subset that maximizes a objective function. The paper discusses combinatorial optimization problems, where for each dimension the set of feasible subsets is…
Understanding how information is represented in neural networks is a fundamental challenge in both neuroscience and artificial intelligence. Despite their nonlinear architectures, recent evidence suggests that neural networks encode…
Combinatorial optimization finds an optimal solution within a discrete set of variables and constraints. The field has seen tremendous progress both in research and industry. With the success of deep learning in the past decade, a recent…
The neural networks have trained on incomplete sets that a doctor could collect. Trained neural networks have correctly classified all the presented instances. The number of intervals entered for encoding the quantitative variables is equal…
We present a comparison study between a cluster and factor graph representation of LDPC codes. In probabilistic graphical models, cluster graphs retain useful dependence between random variables during inference, which are advantageous in…
Variable-length codes are the bases of the free submonoids of a free monoid. There are some important longstanding open questions about the structure of finite maximal codes. In this paper we discuss this conjectures and their relations…
The mapping of lexical meanings to wordforms is a major feature of natural languages. While usage pressures might assign short words to frequent meanings (Zipf's law of abbreviation), the need for a productive and open-ended vocabulary,…
We define a notion of complexity, which quantifies the nonlinearity of the computation of a neural network, as well as a complementary measure of the effective dimension of feature representations. We investigate these observables both for…
In the paper we show the existence of a large class of extended perfect binary codes containing maximum ij-components.
In recent years codes that are not Uniquely Decipherable (UD) are been studied partitioning them in classes that localize the ambiguities of the code. A natural question is how we can extend the notion of maximality to codes that are not…