Input-state-output representations and constructions of finite-support 2D convolutional codes

Two-dimensional convolutional codes are considered, with codewords having compact support indexed in N^2 and taking values in F^n, where F is a finite field. Input-state-output representations of these codes are introduced and several aspects of such representations are discussed. Constructive procedures of such codes with a designed distance are also presented. © 2010 AIMS-SDU

Realization of 2D convolutional codes of rate 1/n by separable Roesser models

In this paper, two-dimensional convolutional codes constituted by sequences in where is a finite field, are considered. In particular, we restrict to codes with rate and we investigate the problem of minimal dimension for realizations of such codes by separable Roesser models. The encoders which allow to obtain such minimal realizations, called R-minimal encoders, are characterized

Series concatenation of 2D convolutional codes

In this paper we study two-dimensional (2D) con-volutional codes which are obtained from series concatenation of two 2D convolutional codes. In this preliminary work we confine ourselves to dealing with finite-support 2D convolutional codes and make use of the so-called Fornasini-Marchesini input-state-output (ISO) model representations. In particular, we show that the series concatenation of two 2D convolutional codes is again a 2D convolutional code and we explicitly compute an ISO representation of the code. Within these ISO representations we study when the structural properties of reachability and observability of the two given ISO representations carry over to the resulting 2D convolutional code

DECODING OF 2-D CONVOLUTIONAL CODES BASED ON ALGEBRAIC APPROACH

In this paper, we apply the decoding matrix for 2-D convolution codes to reconstruct information sequences. It is suitable for non-square matrices with multivariate polynomial elements. Next, development of a syndrome decoder for 2-D convolutional codes based on Gröbner bases is introduced. The computation of the syndrome vector employs the computation of the syzygy module, found by means of the Gröbner basis of a certain module. Then, estimated error vector can be identified by using m-variate division algorithm. Simulation results show error-correcting capability of decoding process

Series concatenation of 2D convolutional codes by means of input-state-output representations

In this paper we investigate the properties of two-dimensional ($2$D) convolutional codes which are obtained from series concatenation of two $2$D convolutional codes. For this purpose we confine ourselves to dealing with finite-support $2$D convolutional codes and make use of the so-called Fornasini-Marchesini input-state-output (ISO) model representations. Within these ISO representations we study when the structural properties of modal reachability and modal observability of the two given ISO representations carry over to the resulting $2$D convolutional code. Moreover, we provide necessary conditions for obtaining a systematic concatenated convolutional code. Finally, we present a lower bound on its free distance.publishe

Hyperbolic Deep Neural Networks: A Survey

Recently, there has been a rising surge of momentum for deep representation learning in hyperbolic spaces due to theirhigh capacity of modeling data like knowledge graphs or synonym hierarchies, possessing hierarchical structure. We refer to the model as hyperbolic deep neural network in this paper. Such a hyperbolic neural architecture potentially leads to drastically compact model withmuch more physical interpretability than its counterpart in Euclidean space. To stimulate future research, this paper presents acoherent and comprehensive review of the literature around the neural components in the construction of hyperbolic deep neuralnetworks, as well as the generalization of the leading deep approaches to the Hyperbolic space. It also presents current applicationsaround various machine learning tasks on several publicly available datasets, together with insightful observations and identifying openquestions and promising future directions

Deep representations of structures in the 3D-world

This thesis demonstrates a collection of neural network tools that leverage the structures and symmetries of the 3D-world. We have explored various aspects of a vision system ranging from relative pose estimation to 3D-part decomposition from 2D images. For any vision system, it is crucially important to understand and to resolve visual ambiguities in 3D arising from imaging methods. This thesis has shown that leveraging prior knowledge about the structures and the symmetries of the 3D-world in neural network architectures brings about better representations for ambiguous situations. It helps solve problems which are inherently ill-posed