arXiv subjects

Sueli I. R. Costa

Publications and source records attributed to Sueli I. R. Costa.

At least 19 recordsLinked to original sources

arXiv2025

Multilevel lattice codes from Hurwitz quaternion integers

This work presents an extension of the Construction $π_A$ lattices proposed in \cite{huang2017construction}, to Hurwitz quaternion integers. This construction is provided by using an isomorphism from a version of the Chinese remainder theorem applied to maximal orders in contrast to natural orders in prior works. Exploiting this map, we analyze the performance of the resulting multilevel lattice codes, highlight via computer simulations their notably reduced computational complexity provided by the multistage decoding. Moreover it is shown that this construction effectively attain the Poltyrev-limit.

cs.IT↗

arXiv2025

Achieving uniform side information gain with multilevel lattice codes over the ring of integers

The index coding problem aims to optimise broadcast communication by taking advantage of receiver-side information to improve transmission efficiency. In this letter, we explore the application of Construction $π_A$ lattices to index coding. We introduce a coding scheme, named \textit{CRT lattice index coding}, using Construction $π_A$ over $\mathbb{Z}$ to address the index coding problem. It is derived an upper bound for side information gain of a CRT lattice index code and conditions for the uniformity of this gain. The efficiency of this approach is shown through theoretical analysis and code design examples.

cs.IT↗

arXiv2024

On Closed-Form Expressions for the Fisher-Rao Distance

The Fisher-Rao distance is the geodesic distance between probability distributions in a statistical manifold equipped with the Fisher metric, which is a natural choice of Riemannian metric on such manifolds. It has recently been applied to supervised and unsupervised problems in machine learning, in various contexts. Finding closed-form expressions for the Fisher-Rao distance is generally a non-trivial task, and those are only available for a few families of probability distributions. In this survey, we collect examples of closed-form expressions for the Fisher-Rao distance of both discrete and continuous distributions, aiming to present them in a unified and accessible language. In doing so, we also: illustrate the relation between negative multinomial distributions and the hyperbolic model, include a few new examples, and write a few more in the standard form of elliptical distributions.

math.ST↗

arXiv2023

On Lattice Constructions D and D' from q-ary Linear Codes

Multilevel lattice codes, such as those associated to Constructions $C$, $\overline{D}$, D and D', have relevant applications in communications. In this paper, we investigate some properties of lattices obtained via Constructions D and D' from $q$-ary linear codes. Connections with Construction A, generator matrices, expressions and bounds for the lattice volume and minimum distances are derived. Extensions of previous results regarding construction and decoding of binary and $p$-ary linear codes ($p$ prime) are also presented.

cs.IT↗

arXiv2022

The Fisher-Rao Loss for Learning under Label Noise

Choosing a suitable loss function is essential when learning by empirical risk minimisation. In many practical cases, the datasets used for training a classifier may contain incorrect labels, which prompts the interest for using loss functions that are inherently robust to label noise. In this paper, we study the Fisher-Rao loss function, which emerges from the Fisher-Rao distance in the statistical manifold of discrete distributions. We derive an upper bound for the performance degradation in the presence of label noise, and analyse the learning speed of this loss. Comparing with other commonly used losses, we argue that the Fisher-Rao loss provides a natural trade-off between robustness and training dynamics. Numerical experiments with synthetic and MNIST datasets illustrate this performance.

cs.LG↗

arXiv2022

Information Properties of a Random Variable Decomposition through Lattices

A full-rank lattice in the Euclidean space is a discrete set formed by all integer linear combinations of a basis. Given a probability distribution on $\mathbb{R}^n$, two operations can be induced by considering the quotient of the space by such a lattice: wrapping and quantization. For a lattice $Λ$, and a fundamental domain $D$ which tiles $\mathbb{R}^n$ through $Λ$, the wrapped distribution over the quotient is obtained by summing the density over each coset, while the quantized distribution over the lattice is defined by integrating over each fundamental domain translation. These operations define wrapped and quantized random variables over $D$ and $Λ$, respectively, which sum up to the original random variable. We investigate information-theoretic properties of this decomposition, such as entropy, mutual information and the Fisher information matrix, and show that it naturally generalizes to the more abstract context of locally compact topological groups.

cs.IT↗

arXiv2021

Constructive Spherical Codes by Hopf Foliations

We present a new systematic approach to constructing spherical codes in dimensions $2^k$, based on Hopf foliations. Using the fact that a sphere $S^{2n-1}$ is foliated by manifolds $S_{\cosη}^{n-1} \times S_{\sinη}^{n-1}$, $η\in[0,π/2]$, we distribute points in dimension $2^k$ via a recursive algorithm from a basic construction in $\mathbb{R}^4$. Our procedure outperforms some current constructive methods in several small-distance regimes and constitutes a compromise between achieving a large number of codewords for a minimum given distance and effective constructiveness with low encoding computational cost. Bounds for the asymptotic density are derived and compared with other constructions. The encoding process has storage complexity $O(n)$ and time complexity $O(n \log n)$. We also propose a sub-optimal decoding procedure, which does not require storing the codebook and has time complexity $O(n \log n)$.

cs.IT↗

arXiv2020

On Communication for Distributed Babai Point Computation

We present a communication-efficient distributed protocol for computing the Babai point, an approximate nearest point for a random vector ${\bf X}\in\mathbb{R}^n$ in a given lattice. We show that the protocol is optimal in the sense that it minimizes the sum rate when the components of ${\bf X}$ are mutually independent. We then investigate the error probability, i.e. the probability that the Babai point does not coincide with the nearest lattice point. In dimensions two and three, this probability is seen to grow with the packing density. For higher dimensions, we use a bound from probability theory to estimate the error probability for some well-known lattices. Our investigations suggest that for uniform distributions, the error probability becomes large with the dimension of the lattice, for lattices with good packing densities. We also consider the case where $\mathbf{X}$ is obtained by adding Gaussian noise to a randomly chosen lattice point. In this case, the error probability goes to zero with the lattice dimension when the noise variance is sufficiently small. In such cases, a distributed algorithm for finding the approximate nearest lattice point is sufficient for finding the nearest lattice point.

cs.IT↗

arXiv2018

On Equivalence of Binary Asymmetric Channels regarding the Maximum Likelihood Decoding

We study the problem of characterizing when two memoryless binary asymmetric channels, described by their transition probabilities $(p,q)$ and $(p',q')$, are equivalent from the point of view of maximum likelihood decoding (MLD) when restricted to $n$-block binary codes. This equivalence of channels induces a partition (depending on $n$) on the space of parameters $(p,q)$ into regions associated with the equivalence classes. Explicit expressions for describing these regions, their number and areas are derived. Some perspectives of applications of our results to decoding problems are also presented.

cs.IT↗

arXiv2018

Communication-Efficient Search for an Approximate Closest Lattice Point

We consider the problem of finding the closest lattice point to a vector in n-dimensional Euclidean space when each component of the vector is available at a distinct node in a network. Our objectives are (i) minimize the communication cost and (ii) obtain the error probability. The approximate closest lattice point considered here is the one obtained using the nearest-plane (Babai) algorithm. Assuming a triangular special basis for the lattice, we develop communication-efficient protocols for computing the approximate lattice point and determine the communication cost for lattices of dimension n>1. Based on available parameterizations of reduced bases, we determine the error probability of the nearest plane algorithm for two dimensional lattices analytically, and present a computational error estimation algorithm in three dimensions. For dimensions 2 and 3, our results show that the error probability increases with the packing density of the lattice.

cs.IT↗

arXiv2016

Bounds for the $l_1$-distance of $q$-ary lattices obtained via Constructions D, D$^{'}$ and $\overline{D}$

Lattices have been used in several problems in coding theory and cryptography. In this paper we approach $q$-ary lattices obtained via Constructions D, $\D'$ and $\overline{D}$. It is shown connections between Constructions D and $\D'$. Bounds for the minimum $l_1$-distance of lattices $Λ_{D}$, $Λ_{D'}$ and $Λ_{\overline{D}}$ and, under certain conditions, a generator matrix for $Λ_{D'}$ are presented. In addition, when the chain of codes used is closed under the zero-one addition, we derive explicit expressions for the minimum $l_1$-distances of the lattices $Λ_{D}$ and $Λ_{\overline{D}}$ attached to the distances of the codes used in these constructions.

cs.IT↗

arXiv2016

Lattices from codes over $\mathbb{Z}_q$: Generalization of Constructions $D$, $D'$ and $\overline{D}$

In this paper, we extend the lattice Constructions $D$, $D'$ and $\overline{D}$ $($this latter is also known as Forney's code formula$)$ from codes over $\mathbb{F}_p$ to linear codes over $\mathbb{Z}_q$, where $q \in \mathbb{N}$. We define an operation in $\mathbb{Z}_q^n$ called zero-one addition, which coincides with the Schur product when restricted to $\mathbb{Z}_2^n$ and show that the extended Construction $\overline{D}$ produces a lattice if and only if the nested codes are closed under this addition. A generalization to the real case of the recently developed Construction $A'$ is also derived and we show that this construction produces a lattice if and only if the corresponding code over $\mathbb{Z}_q[X]/X^a$ is closed under a shifted zero-one addition. One of the motivations for this work is the recent use of $q$-ary lattices in cryptography.

cs.IT↗

arXiv2016

On Cube Tilings of Tori and Classification of Perfect Codes in the Maximum Metric

We describe odd-length-cube tilings of the n-dimensional q-ary torus what includes q-periodic integer lattice tilings of R^n. In the language of coding theory these tilings correspond to perfect codes with respect to the maximum metric. A complete characterization of the two-dimensional tillings is presented and in the linear case, a description of general matrices, isometry and isomorphism classes is provided. Several methods to construct perfect codes from codes of smaller dimension or via sections are derived. We introduce a special type of matrices (perfect matrices) which are in correspondence with generator matrices for linear perfect codes in arbitrary dimensions. For maximal perfect codes, a parametrization obtained allows to describe isomorphism classes of such codes. We also approach the problem of what isomorphism classes of abelian groups can be represented by q-ary n-dimensional perfect codes of a given cardinality N.

math.CO↗

arXiv2015

Perfect codes in the lp metric

We investigate perfect codes in $\mathbb{Z}^n$ under the $\ell_p$ metric. Upper bounds for the packing radius $r$ of a linear perfect code, in terms of the metric parameter $p$ and the dimension $n$ are derived. For $p = 2$ and $n = 2, 3$, we determine all radii for which there are linear perfect codes. The non-existence results for codes in $\mathbb{Z}^n$ presented here imply non-existence results for codes over finite alphabets $\mathbb{Z}_q$, when the alphabet size is large enough, and has implications on some recent constructions of spherical codes.

math.CO↗

arXiv2015

Quasi-perfect codes in the $\ell_p$ metric

We consider quasi-perfect codes in $\mathbb{Z}^n$ over the $\ell_p$ metric, $2 \leq p < \infty$. Through a computational approach, we determine all radii for which there are linear quasi-perfect codes for $p = 2$ and $n = 2, 3$. Moreover, we study codes with a certain \textit{degree of imperfection}, a notion that generalizes the quasi-perfect codes. Numerical results concerning the codes with the smallest degree of imperfection are presented.

cs.IT↗

arXiv2014

Fisher information distance: a geometrical reading

This paper is a strongly geometrical approach to the Fisher distance, which is a measure of dissimilarity between two probability distribution functions. The Fisher distance, as well as other divergence measures, are also used in many applications to establish a proper data average. The main purpose is to widen the range of possible interpretations and relations of the Fisher distance and its associated geometry for the prospective applications. It focuses on statistical models of the normal probability distribution functions and takes advantage of the connection with the classical hyperbolic geometry to derive closed forms for the Fisher distance in several cases. Connections with the well-known Kullback-Leibler divergence measure are also devised.

stat.ME↗

arXiv2013

Curves on Flat Tori and Analog Source-Channel Codes

In this paper we consider the problem of transmitting a continuous alphabet discrete-time source over an AWGN channel. We propose a constructive scheme based on a set of curves on the surface of a N-dimensional sphere. Our approach shows that the design of good codes for this communication problem is related to geometrical properties of spherical codes and projections of N-dimensional rectangular lattices. Theoretical comparisons with some previous works in terms of the mean square error as a function of the channel SNR as well as simulations are provided.

cs.IT↗

arXiv2013

Optimum Commutative Group Codes

A method for finding an optimum $n$-dimensional commutative group code of a given order $M$ is presented. The approach explores the structure of lattices related to these codes and provides a significant reduction in the number of non-isometric cases to be analyzed. The classical factorization of matrices into Hermite and Smith normal forms and also basis reduction of lattices are used to characterize isometric commutative group codes. Several examples of optimum commutative group codes are also presented.

cs.IT↗