Prediction of Multidimensional Spatial Variation Data via Bayesian Tensor Completion

Abstract

This paper presents a multi-dimensional computational method to predict the spatial variation data inside and across multiple dies of a wafer. This technique is based on tensor computation. A tensor is a high-dimensional generalization of a matrix or a vector. By exploiting the hidden low-rank property of a high-dimensional data array, the large amount of unknown variation testing data may be predicted from a few random measurement samples. The tensor rank, which decides the complexity of a tensor representation, is decided by an available variational Bayesian approach. Our approach is validated by a practical chip testing data set, and it can be easily generalized to characterize the process variations of multiple wafers. Our approach is more efficient than the previous virtual probe techniques in terms of memory and computational cost when handling high-dimensional chip testing data.

Tensor Methods for Generating Compact Uncertainty Quantification and Deep Learning Models

“…For instance, in samplingbased techniques, the number of simulation samples may increase exponentially as d increases. Some tensor solvers have been developed to address this fundamental challenge: [38], [39] high-dim stochastic collocation tensor completion to estimate unknown simulation data [37] hierarchical uncertainty quantification tensor-train decomposition for high-dim integration [33] uncertainty analysis with non-Gaussian correlated uncertainty functional tensor train to compute basis functions [40] spatial variation pattern prediction statistical tensor completion to predict variation pattern resulting generalized polynomial chaos expansion is sparse. This technique has been successfully applied to electronic IC, photonics and MEMS with up to 57 random parameters.…”

“…In our recent paper [40], we proposed to simultaneously predict the variation patterns of multiple dies. If each die has N 1 ×N 2 devices to test, we can stack N 3 dies together to form a tensor.…”

Rank-Revealing Block-Term Decomposition for Tensor Completion

“…The ubiquity in our big data era of data with (explicitly or implicitly) multiple dimensions/relations that are often incomplete and/or uncertain has given rise to numerous applications of tensor completion [1], such as image and video in-painting [2], hyperspectral imaging [3], prediction of multi-dimensional non-stationary wireless channels [4] semiconductor manufacturing [5], and computational materials science [6], to name only a few. Though still lacking in sufficient theoretical foundations and algorithmic variety when compared to its matrix-based counterpart, tensor completion has already a quite rich literature [1], which includes (among other approaches) methods based on lowrank decomposition models.…”

Blind Hyperspectral and Multispectral Image Fusion Using Coupled Non-Negative Tucker Tensor Decomposition

“…Noteworthy, stacking a 3D data into matrix form in NMFbased approaches loses the neighborhood structures, smoothness, and continuity characteristics. In this regard, tensors or multiway arrays have been frequently used in multidimensional data analysis [26,28,[34][35][36]. Additionally, exploiting the power of multilinear algebra of the tensor representation shows more flexibility in the choice of constraints that match data properties.…”