Deep neural networks based on state space models (SSMs) are attracting significant attention in sequence modeling since their computational cost is much smaller than that of Transformers. While the capabilities of SSMs have been demonstrated through experiments in various tasks, theoretical understanding of SSMs is still limited. In particular, most theoretical studies discuss the capabilities of SSM layers without nonlinear layers, and there is a lack of discussion on their combination with nonlinear layers. In this paper, we explore the capabilities of SSMs combined with fully connected neural networks, and show that they are comparable to Transformers in extracting the essential tokens depending on the input. As concrete examples, we consider two synthetic tasks, which are challenging for a single SSM layer, and demonstrate that SSMs combined with nonlinear layers can efficiently solve these tasks. Furthermore, we study the nonparametric regression task, and prove that the ability of SSMs is equivalent to that of Transformers in estimating functions belonging to a certain class.

Mixture of Experts (MoE), an ensemble of specialized models equipped with a router that dynamically distributes each input to appropriate experts, has achieved successful results in the field of machine learning. However, theoretical understanding of this architecture is falling behind due to its inherent complexity. In this paper, we theoretically study the sample and runtime complexity of MoE following the stochastic gradient descent when learning a regression task with an underlying cluster structure of single index models. On the one hand, we show that a vanilla neural network fails in detecting such a latent organization as it can only process the problem as a whole. This is intrinsically related to the concept of *information exponent* which is low for each cluster, but increases when we consider the entire task. On the other hand, with a MoE, we show that it succeeds in dividing the problem into easier subproblems by leveraging the ability of each expert to weakly recover the simpler function corresponding to an individual cluster. To the best of our knowledge, this work is among the first to explore the benefits of the MoE framework by examining its SGD dynamics in the context of nonlinear regression.

Pretrained transformers have demonstrated the ability to implement various algorithms at inference time without parameter updates. While theoretical works have established this capability through constructions and approximation guarantees, the optimization and statistical efficiency aspects remain understudied. In this work, we investigate how transformers learn features in-context – a key mechanism underlying their inference-time adaptivity. We focus on the in-context learning of single-index models $y=\sigma_*(⟨\\boldsymbol{x},\\boldsymbol{\beta}⟩)$, which are low-dimensional nonlinear functions parameterized by feature vector $\\boldsymbol\beta$. We prove that transformers pretrained by gradient-based optimization can perform inference-time feature learning, i.e., extract information of the target features $\\boldsymbol{\beta}$ solely from test prompts (despite $\\boldsymbol {\beta}$ varying across different prompts), hence achieving an in-context statistical efficiency that surpasses any non-adaptive (fixed-basis) algorithms such as kernel methods. Moreover, we show that the inference-time sample complexity surpasses the Correlational Statistical Query (CSQ) lower bound, owing to nonlinear label transformations naturally induced by the Softmax self-attention mechanism.

Transformers can implement both generalizable algorithms (e.g., induction heads) and simple positional shortcuts (e.g., memorizing fixed output positions). In this work, we study how the choice of pretraining data distribution steers a shallow transformer toward one behavior or the other. Focusing on a minimal trigger-output prediction task -- copying the token immediately following a special trigger upon its second occurrence -- we present a rigorous analysis of gradient-based training of a single-layer transformer. In both the infinite and finite sample regimes, we prove a transition in the learned mechanism: if input sequences exhibit sufficient diversity, measured by a low “max-sum” ratio of trigger-to-trigger distances, the trained model implements an induction head and generalizes to unseen contexts; by contrast, when this ratio is large, the model resorts to a positional shortcut and fails to generalize out-of-distribution (OOD). We also reveal a trade-off between the pretraining context length and OOD generalization, and derive the optimal pretraining distribution that minimizes computational cost per sample. Finally, we validate our theoretical predictions with controlled synthetic experiments, demonstrating that broadening context distributions robustly induces induction heads and enables OOD generalization. Our results shed light on the algorithmic biases of pretrained transformers and offer conceptual guidelines for data-driven control of their learned behaviors.

Linear attention has attracted interest as a computationally efficient approximation to softmax attention, especially for long sequences. Recent studies has explored distilling softmax attention in pre-trained Transformers into linear attention. However, a critical challenge remains: *how to choose the feature dimension that governs the approximation quality*. Existing methods fix this dimension uniformly across all attention layers, overlooking the diverse roles and complexities of them. In this paper, we propose a principled method to automatically determine the feature dimension in linear attention using the concept of statistical *degrees of freedom*, which represent the effective dimensionality of the inputs. We provide a theoretical bound on the approximation error and show that the dimension chosen by our method achieves smaller errors under a fixed computational budget. Furthermore, we introduce an efficient layerwise training strategy to learn nonlinear features tailored to each layer. Experiments on multiple pre-trained transformers demonstrate that our method improves the performance of distilled models compared to baselines without increasing the inference cost. Our findings also provide insight into how the complexity of the attention mechanism evolves across layers.

Machine learning for point clouds has been attracting much attention, with many applications in various fields, such as shape recognition and material science. For enhancing the accuracy of such machine learning methods, it is often effective to incorporate global topological features, which are typically extracted by persistent homology. In the calculation of persistent homology for a point cloud, we choose a filtration for the point cloud, an increasing sequence of spaces. Since the performance of machine learning methods combined with persistent homology is highly affected by the choice of a filtration, we need to tune it depending on data and tasks. In this paper, we propose a framework that learns a filtration adaptively with the use of neural networks. In order to make the resulting persistent homology isometry-invariant, we develop a neural network architecture with such invariance. Additionally, we show a theoretical result on a finite-dimensional approximation of filtration functions, which justifies the proposed network architecture. Experimental results demonstrated the efficacy of our framework in several classification tasks.

Analysis of neural network optimization in the mean-field regime is important as the setting allows for feature learning. Existing theory has been developed mainly for neural networks in finite dimensions, i.e., each neuron has a finite-dimensional parameter. However, the setting of infinite-dimensional input naturally arises in machine learning problems such as nonparametric functional data analysis and graph classification. In this paper, we develop a new mean-field analysis of two-layer neural network in an infinite-dimensional parameter space. We first give a generalization error bound, which shows that the regularized empirical risk minimizer properly generalizes when the data size is sufficiently large, despite the neurons being infinite-dimensional. Next, we present two gradient-based optimization algorithms for infinite-dimensional mean-field networks, by extending the recently developed particle optimization framework to the infinite-dimensional setting. We show that the proposed algorithms converge to the (regularized) global optimal solution, and moreover, their rates of convergence are of polynomial order in the online setting and exponential order in the finite sample setting, respectively. To our knowledge this is the first quantitative global optimization guarantee of neural network on infinite-dimensional input and in the presence of feature learning.

Accelerator-in-Switch (AiS) is a framework for building an accelerator logic tightly coupled with a switching hub in a single FPGA for high performance computation with heterogeneous environment with CPUs and GPUs. AiS is implemented on a partial reconfigurable region of an FPGA whose permanent region is used for a switching hub. A port of the switching hub is connected to the registers and local memory of AiS directly. AiS has a standard interface for a standard bus (Avalon MM bus, here) to exchange data between on-board DDR SDRAM and the local memory, and various types of accelerators can be implemented just by providing such an interface. The data input and output are performed with the DMA controller inside the switching hub with the shared memory model between host CPUs and GPUs. We implemented two example accelerators: a reduction calculator for a radiation transfer equations solver (RED) and LET generator for N-body simulation (LET) were implemented as the AiS on PEACH3, a switching hub for a PCIe direct interconnection network with Altera's Stratix V. The use of partial reconfiguration makes it possible to switch multiple accelerators without stopping the switching hub. As a result, we reduced the time for place&route of an accelerator by 47% compared to the case of the design combining the accelerator into the switching hub.