Self-consistent Field (SCF) equation is a type of nonlinear eigenvalue problem in which the matrix to be eigen-decomposed is a function of its own eigenvectors. It is of great significance in computational science for its connection to the Schrödinger equation. Traditional fixed-point iteration methods for solving such equations suffer from non-convergence issues. In this work, we present a novel perspective on such SCF equations as a principal component analysis (PCA) for non-stationary time series, in which a distribution and its own top principal components are mutually updated over time, and the equilibrium state of the model corresponds to the solution of the SCF equations. By the new perspective, online PCA techniques are able to engage in so as to enhance the convergence of the model towards the equilibrium state, acting as a new set of tools for converging the SCF equations. With several numerical adaptations, we then develop a new algorithm for converging the SCF equation, and demonstrated its high convergence capacity with experiments on both synthesized and real electronic structure scenarios.

In this section, we briefly introduce the physical background of Equation (8) in Section 3.5 of the

We consider the online Principal Component Analysis (PCA) for contaminated samples (containing outliers) which are revealed sequentially to the Principal Components (PCs) estimator. Due to their sensitiveness to outliers, previous online PCA algorithms fail in this case and their results can be arbitrarily bad. Here we propose the online robust PCA algorithm, which is able to improve the PCs estimation upon an initial one steadily, even when faced with a constant fraction of outliers. We show that the final result of the proposed online RPCA has an acceptable degradation from the optimum. Actually, under mild conditions, online RPCA achieves the maximal robustness with a $50\%$ breakdown point. Moreover, online RPCA is shown to be efficient for both storage and computation, since it need not re-explore the previous samples as in traditional robust PCA algorithms.

Catastrophic forgetting remains a critical challenge in continual learning for large language models (LLMs), where models struggle to retain performance on historical tasks when fine-tuning on new sequential data without access to past datasets. In this paper, we first reveal that the drift of functional directions during the fine-tuning process is a key reason why existing regularization-based methods fail in long-term LLM continual learning. To address this, we propose Dynamic Orthogonal Continual (DOC) fine-tuning, a novel approach that tracks the drift of these functional directions and dynamically updates them during the fine-tuning process. Furthermore, by adjusting the gradients of new task parameters to be orthogonal to the tracked historical function directions, our method mitigates interference between new and old tasks. Extensive experiments on various LLM continual learning benchmarks demonstrate that this approach outperforms prior methods, effectively reducing catastrophic forgetting and providing a robust tool for continuous LLM fine-tuning. Our code is available at https://github.com/meloxxxxxx/DOC.

Self-consistent Field (SCF) equation is a type of nonlinear eigenvalue problem in which the matrix to be eigen-decomposed is a function of its own eigenvectors. It is of great significance in computational science for its connection to the Schrödinger equation. Traditional fixed-point iteration methods for solving such equations suffer from non-convergence issues. In this work, we present a novel perspective on such SCF equations as a principal component analysis (PCA) for non-stationary time series, in which a distribution and its own top principal components are mutually updated over time, and the equilibrium state of the model corresponds to the solution of the SCF equations. By the new perspective, online PCA techniques are able to engage in so as to enhance the convergence of the model towards the equilibrium state, acting as a new set of tools for converging the SCF equations. With several numerical adaptations, we then develop a new algorithm for converging the SCF equation, and demonstrated its high convergence capacity with experiments on both synthesized and real electronic structure scenarios.

This equation, and others, probably overlap quite substantially with the approaches taken in e.g. "Optimization Algorithms on Matrix Manifolds" by Absil, P., Mahony, R., Sepulchre, R. ( this reference also addresses more general gradient flows, stepsize, retraction ("projection"), and convergence issues). S. Bonnabel, "Stochastic gradient descent on Riemannian manifolds" (arXiv) A. Edelman et al. "THE GEOMETRY OF ALGORITHMS WITH ORTHOGONALITY CONSTRAINTS" - The perspective of gradient descent on matrix manifolds is, to this reviewer, a major omission. The paper could greatly benefit from a discussion as to how this slice of the literature fits in to the picture, in terms of both practical algorithms and theory. I understand that space is a constraint. But if the authors decide to ultimately write a journal version of the submission, careful, considered inclusion of the above would make for a fantastic, authoritative paper bridging all of the major points of view on the topic.

We consider the online Principal Component Analysis (PCA) for contaminated samples (containing outliers) which are revealed sequentially to the Principal Components (PCs) estimator. Due to their sensitiveness to outliers, previous online PCA algorithms fail in this case and their results can be arbitrarily bad. Here we propose the online robust PCA algorithm, which is able to improve the PCs estimation upon an initial one steadily, even when faced with a constant fraction of outliers. We show that the final result of the proposed online RPCA has an acceptable degradation from the optimum. Actually, under mild conditions, online RPCA achieves the maximal robustness with a $50\%$ breakdown point.

Non-convex optimization with global convergence guarantees is gaining significant interest in machine learning research in recent years. However, while most works consider either offline settings in which all data is given beforehand, or simple online stochastic i.i.d. settings, very little is known about non-convex optimization for adversarial online learning settings. In this paper we focus on the problem of Online Principal Component Analysis in the regret minimization framework. For this problem, all existing regret minimization algorithms are based on a positive semidefinite convex relaxation, and hence require quadratic memory and SVD computation (either thin of full) on each iteration, which amounts to at least quadratic runtime per iteration. This is in stark contrast to a corresponding stochastic i.i.d. variant of the problem which admits very efficient gradient ascent algorithms that work directly on the natural non-convex formulation of the problem, and hence require only linear memory and linear runtime per iteration. This raises the question: \textit{can non-convex online gradient ascent algorithms be shown to minimize regret in online adversarial settings?} In this paper we take a step forward towards answering this question. We introduce an \textit{adversarially-perturbed spiked-covariance model} in which, each data point is assumed to follow a fixed stochastic distribution, but is then perturbed by adversarial noise. We show that in a certain regime of parameters, when the non-convex online gradient ascent algorithm is initialized with a "warm-start" vector, it provably minimizes the regret with high probability. We further discuss the possibility of computing such a "warm-start" vector. Our theoretical findings are supported by empirical experiments on both synthetic and real-world data.

This paper studies the complexity of the stochastic gradient algorithm for PCA when the data are observed in a streaming setting. We also propose an online approach for selecting the learning rate. Simulation experiments confirm the practical relevance of the plain stochastic gradient approach and that drastic improvements can be achieved by learning the learning rate.