Better Lattice Quantizers Constructed from Complex Integers

Real-valued lattices and complex-valued lattices are mutually convertible, thus we can take advantages of algebraic integers to defined good lattice quantizers in the real-valued domain. In this paper, we adopt complex integers to define generalized checkerboard lattices, especially $\mathcal{E}_{m}$ and $\mathcal{E}_{m}^+$ defined by Eisenstein integers. Using $\mathcal{E}_{m}^+$, we report the best lattice quantizers in dimensions $14$, $18$, $20$, and $22$. Their product lattices with integers $\mathbb{Z}$ also yield better quantizers in dimensions $15$, $19$, $21$, and $23$. The Conway-Sloane type fast decoding algorithms for $\mathcal{E}_{m}$ and $\mathcal{E}_{m}^+$ are given.Comment: 7 page

Faster Projection in Sphere Decoding

Most of the calculations in standard sphere decoders are redundant, in the sense that they either calculate quantities that are never used or calculate some quantities more than once. A new method, which is applicable to lattices as well as finite constellations, is proposed to avoid these redundant calculations while still returning the same result. Pseudocode is given to facilitate immediate implementation. Simulations show that the speed gain with the proposed method increases linearly with the lattice dimension. At dimension 60, the new algorithms avoid about 75% of all floating-point operations

MML Probabilistic Principal Component Analysis

Principal component analysis (PCA) is perhaps the most widely method for data dimensionality reduction. A key question in PCA decomposition of data is deciding how many factors to retain. This manuscript describes a new approach to automatically selecting the number of principal components based on the Bayesian minimum message length method of inductive inference. We also derive a new estimate of the isotropic residual variance and demonstrate, via numerical experiments, that it improves on the usual maximum likelihood approach

The Voronoi Region of the Barnes-Wall Lattice Λ16\Lambda_{16}

We give a detailed description of the Voronoi region of the Barnes-Wall lattice $\Lambda_{16}$, including its vertices, relevant vectors, and symmetry group. The exact value of its quantizer constant is calculated, which was previously only known approximately. To verify the result, we estimate the same constant numerically and propose a new very simple method to quantify the variance of such estimates, which is far more accurate than the commonly used jackknife estimator.Comment: 8 pages, 2 figures. This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessibl

On the best lattice quantizers

A lattice quantizer approximates an arbitrary real-valued source vector with a vector taken from a specific discrete lattice. The quantization error is the difference between the source vector and the lattice vector. In a classic 1996 paper, Zamir and Feder show that the globally optimal lattice quantizer (which minimizes the mean square error) has white quantization noise: for a uniformly distributed source, the covariance of the error is the identity matrix, multiplied by a positive real factor. We generalize the theorem, showing that the same property holds (i) for any locally optimal lattice quantizer and (ii) for an optimal product lattice, if the component lattices are themselves locally optimal. We derive an upper bound on the normalized second moment (NSM) of the optimal lattice in any dimension, by proving that any lower- or upper-triangular modification to the generator matrix of a product lattice reduces the NSM. Using these tools and employing the best currently known lattice quantizers to build product lattices, we construct improved lattice quantizers in dimensions 13 to 15, 17 to 23, and 25 to 48. In some dimensions, these are the first reported lattices with normalized second moments below the Zador upper bound

Complexity and algorithms for computing Voronoi cells of lattices

In this paper we are concerned with finding the vertices of the Voronoi cell of a Euclidean lattice. Given a basis of a lattice, we prove that computing the number of vertices is a #P-hard problem. On the other hand we describe an algorithm for this problem which is especially suited for low dimensional (say dimensions at most 12) and for highly-symmetric lattices. We use our implementation, which drastically outperforms those of current computer algebra systems, to find the vertices of Voronoi cells and quantizer constants of some prominent lattices.Comment: 20 pages, 2 figures, 5 table

UVeQFed: Universal Vector Quantization for Federated Learning

Traditional deep learning models are trained at a centralized server using labeled data samples collected from end devices or users. Such data samples often include private information, which the users may not be willing to share. Federated learning (FL) is an emerging approach to train such learning models without requiring the users to share their possibly private labeled data. In FL, each user trains its copy of the learning model locally. The server then collects the individual updates and aggregates them into a global model. A major challenge that arises in this method is the need of each user to efficiently transmit its learned model over the throughput limited uplink channel. In this work, we tackle this challenge using tools from quantization theory. In particular, we identify the unique characteristics associated with conveying trained models over rate-constrained channels, and propose a suitable quantization scheme for such settings, referred to as universal vector quantization for FL (UVeQFed). We show that combining universal vector quantization methods with FL yields a decentralized training system in which the compression of the trained models induces only a minimum distortion. We then theoretically analyze the distortion, showing that it vanishes as the number of users grows. We also characterize the convergence of models trained with the traditional federated averaging method combined with UVeQFed to the model which minimizes the loss function. Our numerical results demonstrate the gains of UVeQFed over previously proposed methods in terms of both distortion induced in quantization and accuracy of the resulting aggregated model

Local Energy Optimality of Periodic Sets

We study the local optimality of periodic point sets in $\mathbb{R}^n$ for energy minimization in the Gaussian core model, that is, for radial pair potential functions $f_c(r)=e^{-c r}$ with $c>0$. By considering suitable parameter spaces for $m$-periodic sets, we can locally rigorously analyze the energy of point sets, within the family of periodic sets having the same point density. We derive a characterization of periodic point sets being $f_c$-critical for all $c$ in terms of weighted spherical $2$-designs contained in the set. Especially for $2$-periodic sets like the family $\mathsf{D}^+_n$ we obtain expressions for the hessian of the energy function, allowing to certify $f_c$-optimality in certain cases. For odd integers $n\geq 9$ we can hereby in particular show that $\mathsf{D}^+_n$ is locally $f_c$-optimal among periodic sets for all sufficiently large~$c$.Comment: 27 pages, 2 figure

Configurations of sublattices and Dirichlet-Voronoi cells of periodic point sets

Ausgehend vom Gitter-Quantisierungs-Problem behandelt die vorliegende Arbeit zwei geometrisch motivierte Fragestellungen. Zunächst wird die Anzahl der ähnlichen Untergitter eines Gitters L, d.h. Untergitter des Gitters L welche durch Anwendung einer Isometrie und Streckung aus L hervorgehen, zu gegebenem Streckfaktor untersucht. Für ganzzahlige Gitter werden diese unter geeigneten Voraussetzungen (z.B. bei geraden unimodularen Gittern) mit maximal total isotropen Untermoduln regulärer quadratischer Moduln über Restklassenringen der ganzen Zahlen in Bijektion gesetzt. Es wird eine Klassifikation dieser eben genannten Untermoduln, sogar über beliebigen endlichen Ringen, erreicht. Für endliche Hauptidealringe wird diese Klassifikation zur Bestimmung der Anzahlen maximal total isotroper Untergitter benutzt, insbesondere liefert dies in gewissen Fällen die Anzahlen ähnlicher Untergitter ganzzahliger Gitter. Als wichtiges Beispiel dient die Bestimmung der Anzahlen ähnlicher Untergitter des Wurzelgitters E_8. Im Weiteren ermöglicht Voronoi’s zweite Reduktionstheorie, welche quadratische Formen nach ihrer Delone-Zerlegung auf einer zuvor fixierten Punktmenge partitioniert, eine stückweise explizite Berechnung der bekannten Integral-Formel für die Quantisierungs-Konstante, welche auf einen Quotienten eines Polynoms, in den Einträgen einer symmetrischen Matrix, mit einer skalierten Potenz der Determinante dieser Matrix führt. Dies erlaubt es das Quantisierungs-Problem als endliche Sammlung polynomieller Optimierungsprobleme aufzufassen und mit dieser, zumindest stückweise, expliziten Darstellung wird in Dimension 4 schließlich für einige prominente Gitter geklärt ob diese lokale Minima für das Gitter-Quantisierungs-Problem sind