While deep neural networks have achieved remarkable success across a wide range of domains, their underlying mechanisms remain poorly understood, and they are often regarded as black boxes. This gap between empirical performance and theoretical understanding poses a challenge analogous to the pre-axiomatic stage of classical geometry. In this work, we introduce the Pursuit of Subspaces (PoS) hypothesis, an axiomatic framework that formulates neural network behavior through a set of geometric postulates. These axioms, together with their derived consequences, provide a unified perspective on representation, computation, and generalization in both shallow and deep architectures. We show that this framework yields geometric explanations for fundamental questions in deep learning, including representation structure, architectural mechanisms, and generalization behavior, offering a principled step toward a coherent theoretical foundation.

Existing prompt-tuning methods have demonstrated impressive performances in continual learning (CL), by selecting and updating relevant prompts in the vision-transformer models. On the contrary, this paper aims to learn each task by tuning the prompts in the direction orthogonal to the subspace spanned by previous tasks' features, so as to ensure no interference on tasks that have been learned to overcome catastrophic forgetting in CL. However, different from the orthogonal projection in the traditional CNN architecture, the prompt gradient orthogonal projection in the ViT architecture shows completely different and greater challenges, i.e., 1) the high-order and non-linear self-attention operation; 2) the drift of prompt distribution brought by the LayerNorm in the transformer block. Theoretically, we have finally deduced two consistency conditions to achieve the prompt gradient orthogonal projection, which provide a theoretical guarantee of eliminating interference on previously learned knowledge via the self-attention mechanism in visual prompt tuning. In practice, an effective null-space-based approximation solution has been proposed to implement the prompt gradient orthogonal projection. Extensive experimental results demonstrate the effectiveness of anti-forgetting on four class-incremental benchmarks with diverse pre-trained baseline models, and our approach achieves superior performances to state-of-the-art methods.

In this paper, we focus on solving Extensive-F orm Games (EFGs).

One challenge of continual learning is "catastrophic forgetting" [Had+20; VT19; Kem+18]: A model However, if contexts similar to A arise repeatedly, this may be undesirable.. In machine learning, many data sets display cyclic or periodic patterns.

Concept erasure aims to remove specified features from a representation. It can improve fairness (e.g.