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Solving Coupled Tensor Equation $\mathcal{A} \ltimes \mathcal{X} =\mathcal{B}, \ \mathcal{X}\ltimes \mathcal{C}=\mathcal{D}$ using Semi-Tensor Products in the t-product
arXiv:2605.31122v1 Announce Type: new Abstract: This paper investigates the solution of coupled third-order tensor equation $\mathcal{A} \ltimes \mathcal{X} = \mathcal{B},\ \mathcal{X} \ltimes \mathcal{C} = \mathcal{D},$ of arbitrary dimensions by incorporating semi-tensor product (STP) within t-product framework, where the unknown $\mathcal{X}$ can take form of vector, matrix, or tensor. For the unknown $\mathcal{X}$, we establish a necessary and sufficient condition that provides an...
Structural properties of the implicit function defined by an integral self-consistency equation
arXiv:2606.04243v1 Announce Type: new Abstract: We study the integral equation $\int_0^m \eta\rho(\eta)/(C-\eta)\,d\eta = 1$ with $C>m$, where $\rho$ is a $C^1$ probability density on $[0,M]$ vanishing polynomially at $\eta=M$. Setting $\mathcal{I}^+(m) := \lim_{C \downarrow m}\int_0^m \eta\rho(\eta)/(C-\eta)\,d\eta$ and $\Omega := \{m \in (0,M) : \mathcal{I}^+(m) > 1\}$, the equation determines $C$ implicitly as a function of $m$ on $\Omega$, and our object of study is the dimensionless...
Infinite sequences with optimal diaphony, periodic $L_2$-discrepancy, and beyond
arXiv:2606.05482v1 Announce Type: new Abstract: We investigate the periodic $L_2$-discrepancy of infinite sequences $\S_d$ in $[0,1)^d$ and its analytic counterpart, the diaphony. We prove that infinite order-2 digital sequences over $\mathbb{F}_2$ attain the optimal order $L_{2,N}^{{\rm per}}(\S_d) \le C_d (\log N)^{d/2}/N$ for all $N \in \mathbb{N}\setminus \{1\}$, matching known lower bounds for infinitely many $N \in \mathbb{N}$.
Mutually Unbiased Bases for Variational Quantum Initialization: Basis-Union Optimality and Adaptive Family Search
arXiv:2605.16060v2 Announce Type: replace-cross Abstract: We study mutually unbiased bases (MUBs) as structured finite initialization and adaptation families for variational quantum algorithms. The main theoretical result is that, in every dimension admitting a complete set of MUBs, the complete MUB ensemble maximizes isotropic Gaussian random-Hamiltonian width among all unions of d+1 orthonormal bases in C^d. Equivalently, within this basis-union class, it gives the smallest expected...
$M^3$ Scaling Law: Optimizing Multi-Epoch, Multi-Lingual, and Multi-Stage Training for Low-Resource Language Models
arXiv:2410.12325v2 Announce Type: replace Abstract: In this paper, we study a fundamental design problem in pretraining Large Language Models (LLMs) for low-resource language regimes. Existing works adopt multi-epoch, multi-lingual, and multi-stage training to utilize the limited target-language corpus efficiently, but no prior scaling law can compare recipes spanning these approaches under the same compute budget $C$ and target-language corpus size $D_T$, leaving the optimal training setup...
Magenta RealTime 2: Open and Local Live Music Models
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Beyond the Simplex: Balanced Prototype Geometry for Scorer-Agnostic Open-Set Recognition
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An Empirical Audit of Input Encoders for Multi-Channel Signal Transformers
Announce Type: new Abstract: Transformers consuming multi-channel scalar signals must embed $C$ simultaneous values into one $d_{\text{model}}$-dimensional vector per time step. We empirically audit eight input encoders -- spanning a shared-scalar baseline, per-channel linear projections, an orthogonality regulariser, a nonlinear MLP stem, block-partitioned concatenation, channel-independent and channel-as-token architectures, and a projected positional encoding -- on a synthetic benchmark...
The Evaluation Blind Spot: A Stereological Theory of Benchmark Coverage for Large Language Models
arXiv:2606.05169v1 Announce Type: new Abstract: We give a stereological theory of LLM benchmark coverage. For any suite with effective dimensionality d_eff, the visible Hausdorff distance between two convex capability profiles consistent with the same scores is bounded by epsilon + C R m^(-1/(d_eff-1)), with matching Lipschitz lower bound. Empirically, three independent leaderboards (Open LLM v2, an extended 12-benchmark suite, LiveBench) all have d_eff in [2.86, 4.80] on their competitive...
"I've Seen How This Goes": Characterizing Diversity via Progressive Conditional Surprise
arXiv:2606.01811v1 Announce Type: new Abstract: Measuring the diversity of creative outputs is central to evaluating post-training mode collapse, comparing decoding strategies, and quantifying creative behavior in both AI and human writing. We propose a new approach to measuring diversity using in-context learning, of which the ``Decan'' metric, $D_{Ca_n} = C \times a_n$, is the working instance we evaluate: a per-byte score read off the per-token log-probabilities of a base model $\theta$...