Collapse and simplex ETF

Neural collapse [26] is an intuitive observation that happens at the terminal phase of a well-trained model on a balanced dataset that last-layer features converge to within-class mean, and all within-class means and their corresponding classifier vectors converge to ETF as shown in Figure 6. The main results can be concluded as follows: (NC1) Variability of the last-layer features Σ:= Avgi,c{(hic hc)(hic hc)T} collapse within-class: Σ 0, where hic is the last-layer feature of the i-th sample in the c-th class, and hc is the within-class mean of c-th class's features. Last-layer features converge to within-class mean, and all within-class means and their corresponding classifier vectors converge to a simplex ETF. To analyze this phenomenon, some studies simplify deep neural networks as last-layer features and classifier (layer-peeled model)[9, 12, 40, 53] with proper constraints or regularizations. In the view of layer-peeled model (LPM), training W with constraints on the weights can be seen as training the C-class classification head WL = {W1,...,WC} and features H = {h1,...,hN} of all n samples output by last layer of backbone with constraints EW and EH respectively. EH. (6) In the balanced dataset, as described in Lemma 1, any solutions to this model merge neural collapse and form a simplex equiangular tight frame (ETF), which means ETF is optimal classifier in the balanced case of LPM.