Joint Group Invariant Functions on Data-Parameter Domain Induce Universal Neural Networks

The symmetry and geometry of input data are considered to be encoded in the internal data representation inside the neural network, but the specific encoding rule has been less investigated. In this study, we present a systematic method to induce a generalized neural network and its right inverse operator, called the ridgelet transform, from a joint group invariant function on the data-parameter domain. Since the ridgelet transform is an inverse, (1) it can describe the arrangement of parameters for the network to represent a target function, which is understood as the encoding rule, and (2) it implies the universality of the network. Based on the group representation theory, we present a new simple proof of the universality by using Schur's lemma in a unified manner covering a wide class of networks, for example, the original ridgelet transform, formal deep networks, and the dual voice transform. Since traditional universality theorems were demonstrated based on functional analysis, this study sheds light on the group theoretic aspect of the approximation theory, connecting geometric deep learning to abstract harmonic analysis.Comment: NeurReps 202

Deep Ridgelet Transform: Voice with Koopman Operator Proves Universality of Formal Deep Networks

We identify hidden layers inside a deep neural network (DNN) with group actions on the data domain, and formulate a formal deep network as a dual voice transform with respect to the Koopman operator, a linear representation of the group action. Based on the group theoretic arguments, particularly by using Schur's lemma, we show a simple proof of the universality of DNNs.Comment: NeurReps 202

Microstructure Evolution of Carbon Steel by Hot Equal Channel Angular Extrusion

AbstractAn equal channel angular extrusion (ECAE) process equipment which enable a repetitive hot ECAE process without ejecting workpiece with route A and C are developed. This equipment has T-shape 3 actuator axis in horizontal plane and is capable of simulating the formation of fine grained steels in the transformation route. Each actuator (mechanical servo press unit) can be controlled by both position and load with programed motion. The outline of the developed ECAE equipment and the results of preliminary application of the ECAE equipment at an elevated temperature at various pressing speeds ranging from 2 to 32mm/s for a Nb alloyed steel are present. 2 passes via route C at ram speed 16mm/s are also conducted. The ferrite grain size of about 2μm steel is obtained throughout the workpiece at ram speed of 32mm/s, preheated temperature 960oC

Loss of ATF6α in a Human Carcinoma Cell Line Is Compensated not by Its Paralogue ATF6β but by Sustained Activation of the IRE1 and PERK Arms for Tumor Growth in Nude Mice

To survive poor nutritional conditions, tumor cells activate the unfolded protein response, which is composed of the IRE1, PERK, and ATF6 arms, to maintain the homeostasis of the endoplasmic reticulum, where secretory and transmembrane proteins destined for the secretory pathway gain their correct three-dimensional structure. The requirement of the IRE1 and PERK arms for tumor growth in nude mice is established. Here we investigated the requirement for the ATF6 arm, which consists of ubiquitously expressed ATF6α and ATF6β, by constructing ATF6α-knockout (KO), ATF6β-KO, and ATF6α/β-double KO (DKO) in HCT116 cells derived from human colorectal carcinoma. Results showed that these KO cells grew similarly to wild-type (WT) cells in nude mice, contrary to expectations from our analysis of ATF6α-KO, ATF6β-KO, and ATF6α/β-DKO mice. We then found that the loss of ATF6α in HCT116 cells resulted in sustained activation of the IRE1 and PERK arms in marked contrast to mouse embryonic fibroblasts, in which the loss of ATF6α is compensated for by ATF6β. Although IRE1-KO in HCT116 cells unexpectedly did not affect tumor growth in nude mice, IRE1-KO HCT116 cells with ATF6α knockdown grew significantly more slowly than WT or IRE1-KO HCT116 cells. These results have unraveled the situation-dependent differential compensation strategies of ATF6α