A Study of Neural Collapse Phenomenon: Grassmannian Frame, Symmetry, Generalization

In this paper, we extends original Neural Collapse Phenomenon by proving Generalized Neural Collapse hypothesis. We obtain Grassmannian Frame structure from the optimization and generalization of classification. This structure maximally separates features of every two classes on a sphere and does not require a larger feature dimension than the number of classes. Out of curiosity about the symmetry of Grassmannian Frame, we conduct experiments to explore if models with different Grassmannian Frames have different performance. As a result, we discover the Symmetric Generalization phenomenon. We provide a theorem to explain Symmetric Generalization of permutation. However, the question of why different directions of features can lead to such different generalization is still open for future investigation.Comment: 25 pages, 2 figure

Pressure-stabilized divalent ozonide CaO3 and its impact on Earth's oxygen cycles.

High pressure can drastically alter chemical bonding and produce exotic compounds that defy conventional wisdom. Especially significant are compounds pertaining to oxygen cycles inside Earth, which hold key to understanding major geological events that impact the environment essential to life on Earth. Here we report the discovery of pressure-stabilized divalent ozonide CaO3 crystal that exhibits intriguing bonding and oxidation states with profound geological implications. Our computational study identifies a crystalline phase of CaO3 by reaction of CaO and O2 at high pressure and high temperature conditions; ensuing experiments synthesize this rare compound under compression in a diamond anvil cell with laser heating. High-pressure x-ray diffraction data show that CaO3 crystal forms at 35 GPa and persists down to 20 GPa on decompression. Analysis of charge states reveals a formal oxidation state of -2 for ozone anions in CaO3. These findings unravel the ozonide chemistry at high pressure and offer insights for elucidating prominent seismic anomalies and oxygen cycles in Earth's interior. We further predict multiple reactions producing CaO3 by geologically abundant mineral precursors at various depths in Earth's mantle

Towards Decision-Friendly AUC: Learning Multi-Classifier with AUCµ

Area Under the ROC Curve (AUC) is a widely used ranking metric in imbalanced learning due to its insensitivity to label distributions. As a well-known multiclass extension of AUC, Multiclass AUC (MAUC, a.k.a. M-metric) measures the average AUC of multiple binary classifiers. In this paper, we argue that simply optimizing MAUC is far from enough for imbalanced multi-classification. More precisely, MAUC only focuses on learning scoring functions via ranking optimization, while leaving the decision process unconsidered. Therefore, scoring functions being able to make good decisions might suffer from low performance in terms of MAUC. To overcome this issue, we turn to explore AUCµ, another multiclass variant of AUC, which further takes the decision process into consideration. Motivated by this fact, we propose a surrogate risk optimization framework to improve model performance from the perspective of AUCµ. Practically, we propose a two-stage training framework for multi-classification, where at the first stage a scoring function is learned maximizing AUCµ, and at the second stage we seek for a decision function to improve the F1-metric via our proposed soft F1. Theoretically, we first provide sufficient conditions that optimizing the surrogate losses could lead to the Bayes optimal scoring function. Afterward, we show that the proposed surrogate risk enjoys a generalization bound in order of O(1/√N). Experimental results on four benchmark datasets demonstrate the effectiveness of our proposed method in both AUCµ and F1-metric

Thermally Stable Cellulose Nanocrystals toward High-Performance 2D and 3D Nanostructures

Cellulose nanomaterials have attracted much attention in a broad range of fields such as flexible electronics, tissue engineering, and 3D printing for their excellent mechanical strength and intriguing optical properties. Economic, sustainable, and eco-friendly production of cellulose nanomaterials with high thermal stability, however, remains a tremendous challenge. Here versatile cellulose nanocrystals (DM-OA-CNCs) are prepared through fully recyclable oxalic acid (OA) hydrolysis along with disk-milling (DM) pretreatment of bleached kraft eucalyptus pulp. Compared with the commonly used cellulose nanocrystals from sulfuric acid hydrolysis, DM-OA-CNCs show several advantages including large aspect ratio, carboxylated surface, and excellent thermal stability along with high yield. We also successfully demonstrate the fabrication of high-performance films and 3D-printed patterns using DM-OA-CNCs. The high-performance films with high transparency, ultralow haze, and excellent thermal stability have the great potential for applications in flexible electronic devices. The 3D-printed patterns with porous structures can be potentially applied in the field of tissue engineering as scaffolds