Fourier

The z-transform of a sequence is a classical tool used within signal processing, control theory, computer science, and electrical engineering. It allows for studying sequences from their generating functions, with many operations that can be equivalently defined on the original sequence and its $z$-transform. In particular, the z-transform method focuses on asymptotic behaviors and allows the use of Taylor expansions. We present a sequence of results of increasing significance and difficulty for linear models and optimization algorithms, demonstrating the effectiveness and versatility of the z-transform method in deriving new asymptotic results. Starting from the simplest gradient descent iterations in an infinite-dimensional Hilbert space, we show how the spectral dimension characterizes the convergence behavior. We then extend the analysis to Nesterov acceleration, averaging techniques, and stochastic gradient descent.

The architecture of a neural network and the choice of its activation function are both fundamental to its performance. Equally important is ensuring that these two elements are well matched, as their alignment is key to effective representation and learning. In this paper, we introduce the Fourier Multi-Component and Multi-Layer Neural Network (FMMNN), a model that combines sine-type activations with the multi-component and multi-layer structure of MMNNs. In an FMMNN, each component is represented as a trainable linear combination of fixed random sine-type basis functions, while multi-layer composition generates more complex and adaptive high-frequency features. We establish that FMMNNs retain exponential expressive power for function approximation even under a low-rank architectural structure. We also analyze the optimization landscape of FMMNNs and find it to be substantially more favorable than that of standard fully connected neural networks, especially for high-frequency targets. In addition, we propose a scaled random initialization method for the first-layer weights in FMMNNs, which accelerates training and improves final performance when sufficient samples are available. Extensive numerical experiments support our theoretical insights, showing that FMMNNs achieve strong accuracy and favorable convergence behavior on oscillatory function-approximation benchmarks.

In this paper, we study the problem of learning one-dimensional Gaussian mixture models (GMMs) with a specific focus on estimating both the model order and the mixing distribution from independent and identically distributed (i.i.d.) samples. This paper establishes the optimal sampling complexity for model order estimation in one-dimensional Gaussian mixture models. We prove a fundamental lower bound on the number of samples required to correctly identify the number of components with high probability, showing that this limit depends critically on the separation between component means and the total number of components. We then propose a Fourier-based approach to estimate both the model order and the mixing distribution. Our algorithm utilizes Fourier measurements constructed from the samples, and our analysis demonstrates that its sample complexity matches the established lower bound, thereby confirming its optimality. Numerical experiments further show that our method outperforms conventional techniques in terms of efficiency and accuracy.

The NVIDIA Blackwell Ultra (B300) GPU cuts FP64 vector throughput $\sim 30\times$ while multiplying FP8 tensor throughput. After the recovery of FP64 GEMM via Ozaki Scheme II on FP8 tensor cores and the Tensor-Memory Equilibrium model of the companions ("FP8 is All You Need, Part 1" and "Ozaki 2.5") we ask whether the fifth canonical HPC primitive, the full-FP64 $1024^3$ 3-D FFT, can be carried by the same substrate, and answer with a design and its limit. It is a Bailey six-step transform with no FP64 arithmetic: FP8-tensor DFT GEMMs with fused twiddles, residue-domain Karatsuba combines and exact CRT reconstruction whose bulk is a small GEMM on the FP16 tensor path and whose remainder is a Kulisch fixed-point accumulation with a two-sided modulo-$M$ lift, so the only rounding is the final conversion; constants are machine-generated and verified bit-exactly. The central finding: the binding resource is not floating point but a per-output integer epilogue with floor $(c_{\rm epi}/8),B_{\rm mem}$, $c_{\rm epi} \approx 203$-$281$ instructions per output: on B300 it holds the transform at 63-87 ms against a 12.9 ms roof ($4.9$-$6.7\times$ short); at most $1.3$-$1.9\times$ faster than the collapsed native path, possibly no faster at realised issue rates; no software route reaches the roof; on the NVIDIA Rubin GPU emulation loses $8$-$11\times$. An FP32 variant meets the same wall: the cause is per-scalar reconstruction, not FP64. Each floor term names its remedy: the NVIDIA B200 GPU's INT8 tensor core restored with a position-weighted cross-column accumulation primitive, a load-path deconstruction datapath shared with the companions, two ISA idioms and modular reduction at the MMA output give 16.0-23.5 ms with minor hardware and 12.9-15.0 ms with one moderate ask. All figures are projected floors, not measurements, with sensitivities and the FP8 layout condition given.

Single image super-resolution aims to reconstruct high-resolution images from low-resolution inputs. This paper proposes FreeTransformSR, a novel lightweight super-resolution network based on a channel-wise free low-rank learnable transform. The transform learns task-adaptive basis functions in a data-driven manner, enabling adaptive feature modulation with minimal parameter overhead. To further enhance high-frequency detail recovery, we introduce a local feature modulation branch that complements transform-domain processing with depthwise convolution. In addition, a soft complexity adaptive module dynamically fuses the outputs of local convolution and window self-attention branches through a lightweight gating network, adaptively adjusting the fusion ratio based on regional texture characteristics. An adaptive intensity modulation strategy is also incorporated to adjust transform-domain response strength at the sample level, enabling the network to dynamically adjust processing intensity according to input features. Extensive experiments on five benchmark datasets demonstrate that FreeTransformSR achieves competitive PSNR/SSIM performance with significantly fewer parameters and FLOPs. Specifically, FreeTransformSR achieves 32.41 dB on BSD100 x2 and 27.00 dB on Urban100 x4 with only 595K parameters, while delivering faster inference speed than competing methods, making it well-suited for deployment in resource-constrained scenarios. Source code is available at: https://github.com/HJiLi/FreeTransformSR.