Wolkowicz-Styan Upper Bound on the Hessian Eigenspectrum for Cross-Entropy Loss in Nonlinear Smooth Neural Networks

Neural networks (NNs) are central to modern machine learning and achieve state-of-the-art results in many applications. However, the relationship between loss geometry and generalization is still not well understood. The local geometry of the loss function near a critical point is well-approximated by its quadratic form, obtained through a second-order Taylor expansion. The coefficients of the quadratic term correspond to the Hessian matrix, whose eigenspectrum allows us to evaluate the sharpness of the loss at the critical point. Extensive research suggests flat critical points generalize better, while sharp ones lead to higher generalization error. However, sharpness requires the Hessian eigenspectrum, but general matrix characteristic equations have no closed-form solution. Therefore, most existing studies on evaluating loss sharpness rely on numerical approximation methods. Existing closed-form analyses of the eigenspectrum are primarily limited to simplified architectures, such as linear or ReLU-activated networks; consequently, theoretical analysis of smooth nonlinear multilayer neural networks remains limited. Against this background, this study focuses on nonlinear, smooth multilayer neural networks and derives a closed-form upper bound for the maximum eigenvalue of the Hessian with respect to the cross-entropy loss by leveraging the Wolkowicz-Styan bound. Specifically, the derived upper bound is expressed as a function of the affine transformation parameters, hidden layer dimensions, and the degree of orthogonality among the training samples. The primary contribution of this paper is an analytical characterization of loss sharpness in smooth nonlinear multilayer neural networks via a closed-form expression, avoiding explicit numerical eigenspectrum computation. We hope that this work provides a small yet meaningful step toward unraveling the mysteries of deep learning.

Autoencoder-Based Parameter Estimation for Superposed Multi-Component Damped Sinusoidal Signals

Damped sinusoidal oscillations are widely observed in many physical systems, and their analysis provides access to underlying physical properties. However, parameter estimation becomes difficult when the signal decays rapidly, multiple components are superposed, and observational noise is present. In this study, we develop an autoencoder-based method that uses the latent space to estimate the frequency, phase, decay time, and amplitude of each component in noisy multi-component damped sinusoidal signals. We investigate multi-component cases under Gaussian-distribution training and further examine the effect of the training-data distribution through comparisons between Gaussian and uniform training. The performance is evaluated through waveform reconstruction and parameter-estimation accuracy. We find that the proposed method can estimate the parameters with high accuracy even in challenging setups, such as those involving a subdominant component or nearly opposite-phase components, while remaining reasonably robust when the training distribution is less informative. This demonstrates its potential as a tool for analyzing short-duration, noisy signals.

FG-SSA: Features Gradient-based Signals Selection Algorithm of Linear Complexity for Convolutional Neural Networks

Recently, many convolutional neural networks (CNNs) for classification by time domain data of multisignals have been developed. Although some signals are important for correct classification, others are not. When data that do not include important signals for classification are taken as the CNN input layer, the calculation, memory, and data collection costs increase. Therefore, identifying and eliminating nonimportant signals from the input layer are important. In this study, we proposed features gradient-based signals selection algorithm (FG-SSA), which can be used for finding and removing nonimportant signals for classification by utilizing features gradient obtained by the calculation process of grad-CAM. When we define N as the number of signals, the computational complexity of the proposed algorithm is linear time O(N), that is, it has a low calculation cost. We verified the effectiveness of the algorithm using the OPPORTUNITY Activity Recognition dataset, which is an open dataset comprising acceleration signals of human activities. In addition, we checked the average 6.55 signals from a total of 15 acceleration signals (five triaxial sensors) that were removed by FG-SSA while maintaining high generalization scores of classification. Therefore, the proposed algorithm FG-SSA has an effect on finding and removing signals that are not important for CNN-based classification.

Training Process of Unsupervised Learning Architecture for Gravity Spy Dataset

Transient noise appearing in the data from gravitational-wave detectors frequently causes problems, such as instability of the detectors and overlapping or mimicking gravitational-wave signals. Because transient noise is considered to be associated with the environment and instrument, its classification would help to understand its origin and improve the detector's performance. In a previous study, an architecture for classifying transient noise using a time-frequency 2D image (spectrogram) is proposed, which uses unsupervised deep learning combined with variational autoencoder and invariant information clustering. The proposed unsupervised-learning architecture is applied to the Gravity Spy dataset, which consists of Advanced Laser Interferometer Gravitational-Wave Observatory (Advanced LIGO) transient noises with their associated metadata to discuss the potential for online or offline data analysis. In this study, focused on the Gravity Spy dataset, the training process of unsupervised-learning architecture of the previous study is examined and reported.