REKONSTRUKCJA NIEKOMPLETNYCH OBRAZÓW ZA POMOCĄ METOD APROKSYMACJI MODELAMI NISKIEGO RZĘDU

The paper is concerned with the task of reconstructing missing pixels in images perturbed with impulse noise in a transmission channel. Such a task can be formulated in the context of image interpolation on an irregular grid or by approximating an incomplete image by low-rank factor decomposition models. We compared four algorithms that are based on the low-rank decomposition model: SVT, SmNMF-MC , FCSA-TC and SPC-QV. The numerical experiments are carried out for various cases of incomplete images, obtained by removing random pixels or regular grid lines from test images. The best performance is obtained if nonnegativity and smoothing constraints are imposed onto the estimated low-rank factors.W pracy badano zadanie rekonstrukcji brakujących pikseli w obrazach poddanych losowym zaburzeniom impulsowym w kanale transmisyjnym. Takie zadanie może być sformułowane w kontekście interpolacji obrazu na nieregularnej siatce lub aproksymacji niekompletnego obrazu za pomocą modeli dekompozycji obrazu na faktory niskiego rzędu. Porównano skuteczność czterech algorytmów opartych na dekompozycjach macierzy lub tensorów: SVT, SmNMF-MC, FCSA-TC i SPC-QV. Badania przeprowadzono na obrazach niekompletnych, otrzymanych z obrazów oryginalnych przez usunięcie losowo wybranych pikseli lub linii tworzących regularną siatkę. Najwyższą efektywność rekonstrukcji obrazu uzyskano gdy na estymowane faktory niskiego rzędu narzucano ograniczenia nieujemności i gładkości w postaci wagowej filtracji uśredniającej

Exploring Numerical Priors for Low-Rank Tensor Completion with Generalized CP Decomposition

Tensor completion is important to many areas such as computer vision, data analysis, and signal processing. Enforcing low-rank structures on completed tensors, a category of methods known as low-rank tensor completion has recently been studied extensively. While such methods attained great success, none considered exploiting numerical priors of tensor elements. Ignoring numerical priors causes loss of important information regarding the data, and therefore prevents the algorithms from reaching optimal accuracy. This work attempts to construct a new methodological framework called GCDTC (Generalized CP Decomposition Tensor Completion) for leveraging numerical priors and achieving higher accuracy in tensor completion. In this newly introduced framework, a generalized form of CP Decomposition is applied to low-rank tensor completion. This paper also proposes an algorithm known as SPTC (Smooth Poisson Tensor Completion) for nonnegative integer tensor completion as an instantiation of the GCDTC framework. A series of experiments on real-world data indicated that SPTC could produce results superior in completion accuracy to current state-of-the-arts.Comment: 11 pages, 4 figures, 3 pseudocode algorithms, and 1 tabl

Tensor Completion for Weakly-dependent Data on Graph for Metro Passenger Flow Prediction

Low-rank tensor decomposition and completion have attracted significant interest from academia given the ubiquity of tensor data. However, the low-rank structure is a global property, which will not be fulfilled when the data presents complex and weak dependencies given specific graph structures. One particular application that motivates this study is the spatiotemporal data analysis. As shown in the preliminary study, weakly dependencies can worsen the low-rank tensor completion performance. In this paper, we propose a novel low-rank CANDECOMP / PARAFAC (CP) tensor decomposition and completion framework by introducing the $L_{1}$-norm penalty and Graph Laplacian penalty to model the weakly dependency on graph. We further propose an efficient optimization algorithm based on the Block Coordinate Descent for efficient estimation. A case study based on the metro passenger flow data in Hong Kong is conducted to demonstrate improved performance over the regular tensor completion methods.Comment: Accepted at AAAI 202

Nonlinear System Identification via Tensor Completion

Function approximation from input and output data pairs constitutes a fundamental problem in supervised learning. Deep neural networks are currently the most popular method for learning to mimic the input-output relationship of a general nonlinear system, as they have proven to be very effective in approximating complex highly nonlinear functions. In this work, we show that identifying a general nonlinear function $y = f(x_1,\ldots,x_N)$ from input-output examples can be formulated as a tensor completion problem and under certain conditions provably correct nonlinear system identification is possible. Specifically, we model the interactions between the $N$ input variables and the scalar output of a system by a single $N$-way tensor, and setup a weighted low-rank tensor completion problem with smoothness regularization which we tackle using a block coordinate descent algorithm. We extend our method to the multi-output setting and the case of partially observed data, which cannot be readily handled by neural networks. Finally, we demonstrate the effectiveness of the approach using several regression tasks including some standard benchmarks and a challenging student grade prediction task.Comment: AAAI 202

SVDinsTN: An Integrated Method for Tensor Network Representation with Efficient Structure Search

Tensor network (TN) representation is a powerful technique for data analysis and machine learning. It practically involves a challenging TN structure search (TN-SS) problem, which aims to search for the optimal structure to achieve a compact representation. Existing TN-SS methods mainly adopt a bi-level optimization method that leads to excessive computational costs due to repeated structure evaluations. To address this issue, we propose an efficient integrated (single-level) method named SVD-inspired TN decomposition (SVDinsTN), eliminating the need for repeated tedious structure evaluation. By inserting a diagonal factor for each edge of the fully-connected TN, we calculate TN cores and diagonal factors simultaneously, with factor sparsity revealing the most compact TN structure. Experimental results on real-world data demonstrate that SVDinsTN achieves approximately $10^2\sim{}10^3$ times acceleration in runtime compared to the existing TN-SS methods while maintaining a comparable level of representation ability