frame design

Designing wavelet filter banks for arbitrary dimensions and integer dilation matrices remains challenging due to the stringent and often infeasible mixed unitary extension principle (MUEP). Method: This paper introduces the “sum-of-vanishing-products” (SVP) condition as an equivalent yet more tractable alternative to MUEP—rigorously proving their equivalence—and incorporates extended Laplacian pyramid matrices into wavelet frame construction for the first time. Leveraging polynomial sum-of-squares representations, matrix algebra, and multiresolution analysis, we formulate an optimization model for compactly supported tight wavelet frames satisfying SVP constraints. Results: The proposed method substantially reduces design complexity for high-dimensional, arbitrarily dilated tight frames while ensuring numerical stability. Extensive multidimensional numerical experiments validate its effectiveness, flexibility, and universality. The framework provides a tunable, robust, and scalable tool for multiscale geometric analysis.

Traditional discrete representations suffer from resolution dependency, modality coupling, and poor generalization in data reconstruction. To address these limitations, this paper establishes a unified framework for continuous representation (CR), which maps spatial coordinates to continuous functions—enabling resolution-agnostic modeling for tasks such as image reconstruction and novel-view synthesis. Methodologically, we systematically formalize the CR paradigm along three dimensions: algorithmic design, theoretical foundations, and cross-domain applications—constituting the first comprehensive taxonomy. We identify and characterize three core properties: implicit regularization, cross-modal adaptability, and controllable approximation error. The framework encompasses basis-function expansions, statistical modeling, tensor decomposition, and implicit neural representations, supported by convergence proofs and generalization bounds. Furthermore, we release Continuous-Representation-Zoo, an open-source knowledge repository spanning computer vision, graphics, bioinformatics, and remote sensing—advancing the systematic development of continuous representation research.

This work addresses the challenge of unifying diverse structured sparsity patterns for efficient model compression and acceleration. The authors propose S³, an algebraic framework that formally integrates three core components—View (tensor reshaping), Block (atomic pruning units), and Scope (sparsity decision range)—to express a wide spectrum of sparsity patterns, ranging from fine-grained N:M sparsity to coarse-grained channel pruning, within a single formalism. Notably, S³ enables cross-tensor collaborative sparsification. Building upon this framework, the authors incorporate Optimal Brain Damage and Surgeon algorithms to develop structured variants of OBS/OBD. These methods significantly outperform current state-of-the-art second-order heuristic approaches in terms of output reconstruction accuracy.

This work addresses the challenge that conventional graph signal processing methods struggle to effectively model node data in heterogeneous networks due to disparities in dimensionality, modality, and geometric structure. To overcome this limitation, the authors propose a unified framework termed Layered Signal Processing (SSP), which characterizes heterogeneous local signal spaces through network layers and the linear mappings between them, thereby generalizing fundamental operations such as spectral analysis, filtering, and sampling. Key contributions include the first formal definition of the Layered Fourier Transform (SFT), whose frequency basis is constructed from topological and restriction mappings; the introduction of representation layers that accommodate diverse bases, dictionaries, or embeddings while preserving spectral properties; and the design of polynomial layered filters along with a joint node-component sampling strategy. Experiments on synthetic, motion capture, and financial datasets demonstrate significant performance gains over classical baselines, and the framework establishes conditions for perfect reconstruction of bandlimited signals.

This work addresses the problem of reliable extrapolation of list functions under unknown input perturbations—e.g., arbitrary element deletions. Existing methods lack structural constraints, leading to uncontrolled and unreliable extrapolation behavior. To resolve this, we introduce the novel class of *filter-equivariant functions*, which enforce behavioral consistency across all sublists (i.e., outputs on any sublist must align with the restriction of the full-list output). Theoretically, we establish an equivalence between filter-equivariant functions and mapping-equivariant functions, revealing their natural correspondence to a specific class of simplicial complexes—a geometric characterization. Algorithmically, we design a *complete extrapolation algorithm* that exactly reconstructs the output of any filter-equivariant function under arbitrary sublist inputs. Our framework unifies functional programming (via filtering), equivariance theory, combinatorial semantics, and geometric representation, yielding the first formal and computationally tractable theory for structured function extrapolation.

This work investigates the design of pooling-free scattering networks employing fixed monomial nonlinearities to maximize separability for data with low intrinsic dimensionality. By integrating frame theory, geometric measure theory, and moment analysis, the study provides the first geometric characterization of a scattering network’s separation capacity, establishing theoretical bounds for feature extractors operating on low-dimensional rectifiable data. The core contribution consists of two practical design principles: the network’s filters must span a sufficiently broad frequency range, and the frame formed by these filters—when coupled with the data’s geometric structure through a coupling matrix—must exhibit a well-conditioned condition number. These criteria jointly ensure significantly enhanced separation performance, offering concrete guidance for the construction of effective scattering architectures tailored to geometrically structured low-dimensional data.

Traditional image compression relies on fixed orthogonal bases such as the discrete cosine transform (DCT), which are ill-suited to the underlying data distribution and thus limit compression efficiency. This work proposes a trainable isometric multilinear basis, introducing for the first time tensor networks from quantum many-body theory into image compression. By employing block-wise multilinear transforms optimized via Riemannian optimization over the Stiefel manifold of unitary matrices, the method achieves data-adaptive basis learning while preserving near-linear computational complexity, exact invertibility, and an extremely low parameter count. Experiments demonstrate consistent superiority over DCT on both natural images and the Quick Draw dataset; compared to JPEG’s 8×8 DCT, the proposed approach reduces storage requirements by approximately 20% at equivalent reconstruction quality.

This study addresses the challenge that existing orthogonal multiwavelets struggle to simultaneously achieve ultra-compact support, symmetry, and high regularity. Building upon the matrix product filter structure of CL multiwavelets and leveraging the fast Bauer matrix spectral factorization method, this work constructs two novel classes of orthogonal multiwavelets. The proposed filters exhibit orthogonality, symmetry or antisymmetry, and ultra-compact support, with one class demonstrating superior coding efficiency and regularity compared to existing designs. Experimental results show that the new multiwavelets significantly outperform classical counterparts—including GHM, SA4, CL, Integer Haar, and Alpert multiwavelets—in image compression and denoising tasks, as measured by SSIM, MS-SSIM, and human visual perception metrics.