arXiv:2609. 11606v1 Announce Type: cross Abstract: Identifiability of tensor decompositions is often established through linear-algebraic conditions on the factor families.
The paper introduces a new concept called positive scattering to enhance identifiability of nonnegative tensor decompositions. By combining this scattering term with existing dimension-based conditions, the authors derive two sufficient criteria that guarantee minimality, nonnegative rank, and uniqueness for subsets of components. The key result is a positive splitting inequality that links dimension constraints with support-induced geometric rigidity, and the authors show that the scattering term’s mode costs are discrete, enabling an exact activation characterization via graph connectivity. This criterion can certify sparse nonnegative tensor decompositions that elude traditional Kruskal and Lovitz–Petrov conditions, even after reshaping, and reduces to familiar matrix results in the two-dimensional case.
arXiv:2608.28150v2 Announce Type: replace Abstract: How much matrix rank is required to preserve every bounded value output of normalized softmax attention? We study the unrestricted maximum-row-\(\e...
arXiv:2607. 13511v1 Announce Type: cross Abstract: We introduce ExTernD (Expanded-rank Ternary Decomposition), a post-training factorization of each LLM weight matrix $A \in \mathbb{R}^{m \times n}$ into $A \approx B \mathrm{diag}(D) C$ with ternary factors $B \in \{-1,0,+1\}^{m \times k}$, $C \in \{-1,0,+1\}^{k \times n}$ and a real scale vector $D \in \mathbb{R}^k$.
The paper proposes a new dimension‑reduction strategy for transfer‑operator models of dynamical systems that focuses on scoring the σ‑algebra generated by coordinates rather than the operator’s spectral span. By using a χ²‑divergence criterion between embedded present and future states, the method guarantees that twice the intrinsic system dimension suffices to capture the full operator spectrum, even for systems with weakly interacting components that would otherwise require exponentially many modes. Experiments on benchmark systems show that this algebraic approach recovers masked components missed by rank‑based methods and enables accurate prediction of those components from few labels.
arXiv:2605. 25303v3 Announce Type: replace-cross Abstract: The $2 \rightarrow q$ norm of a matrix $X \in \mathbb{R}^{n \times d}$ is defined as $\lVert X \rVert_{2 \rightarrow q} = \sup_{\lVert v \rVert_2 = 1} \lVert Xv \rVert_q$.
arXiv:2609.27836v1 Announce Type: cross Abstract: Let $P$ be an irreducible reversible Markov kernel on a $m$-state space $\Omega$, and denote its right spectral gap $\gamma=1-\lambda_2(P)$. From a s...
arXiv:2602. 04795v3 Announce Type: replace Abstract: Nonnegative matrix factorization (NMF) is a popular data embedding technique.
arXiv:1312. 0925v4 Announce Type: replace Abstract: Alternating Minimization is a widely used and empirically successful heuristic for matrix completion and related low-rank optimization problems.
The paper addresses the limitations of spectral dimension reduction for dynamical systems composed of weakly interacting components, where standard rank‑based methods either require exponentially many modes or omit entire components (a phenomenon termed linear masking). It proposes scoring the σ‑algebra generated by coordinates instead of individual modes, using a χ²‑divergence criterion that guarantees an embedding with twice the intrinsic dimension captures the full operator spectrum. Experiments on benchmark systems show that this algebraic approach recovers masked components and enables accurate prediction from few labels, outperforming traditional rank‑based and VAMP methods.
arXiv:2602. 05869v2 Announce Type: replace-cross Abstract: We introduce Wedge Sampling, a new non-adaptive sampling scheme for low-rank tensor completion.
arXiv:2605. 20440v2 Announce Type: replace Abstract: Symmetry is central to the physical sciences, yet machine learning usually captures it only approximately, leaving a residual per-step equivariance error $\varepsilon$ that compounds with depth $M$ as $M\varepsilon$, whereas exact equivariance holds at unbounded depth; we demonstrate this divergence at fourteen orders of magnitude.