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ARXIV:2603.14578 · THEORETICAL ANALYSIS · SUBMITTED 02 APR · 02:30 UTC · FRESHNESS STALE
ARXIV:2603.14578THEORETICAL ANALYSISSUBMITTED 02 APR · 02:30 UTCFRESHNESS STALEarXiv
This paper explores the spectral structure of data covariance in neural networks through the random feature model.
Opportunity summary
Pain This paper explores the spectral structure of data covariance in neural networks through the random feature model.
Evidence 0 refs | 0 sources | 17% coverage
Blocker Evidence unverified
This paper explores the spectral structure of data covariance in neural networks through the random feature model. A central question is whether this spectral structure is preserved or destroyed when data passes through the…
Scaling laws for neural networks, in which the loss decays as a power-law in the number of parameters, data, and compute, depend fundamentally on the spectral structure of the data covariance, with power-law eigenvalue…
ScienceToStartup currently rates this 2.0/10 on the public viability pass. The proof combines a dyadic head-tail decomposition with Wick chaos expansions for higher-order monomials and random matrix concentration inequalities.
Theoretical Analysis moved forward this cycle; last verified April 2026. Public score 2.0/10.
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This paper explores the spectral structure of data covariance in neural networks through the random feature model.
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10.48550/arXiv.2603.14578This paper explores the spectral structure of data covariance in neural networks through the random feature model.
Abstract
Scaling laws for neural networks, in which the loss decays as a power-law in the number of parameters, data, and compute, depend fundamentally on the spectral structure of the data covariance, with power-law eigenvalue decay appearing ubiquitously in vision and language tasks. A central question is whether this spectral structure is preserved or destroyed when data passes through the basic building block of a neural network: a random linear projection followed by a nonlinear activation. We study this question for the random feature model: given data $x \sim N(0,H)\in \mathbb{R}^v$ where $H$ has $α$-power-law spectrum ($λ_j(H ) \asymp j^{-α}$, $α> 1$), a Gaussian sketch matrix $W \in \mathbb{R}^{v\times d}$, and an entrywise monomial $f(y) = y^{p}$, we characterize the eigenvalues of the population random-feature covariance $\mathbb{E}_{x }[\frac{1}{d}f(W^\top x )^{\otimes 2}]$. We prove matching upper and lower bounds: for all $1 \leq j \leq c_1 d \log^{-(p+1)}(d)$, the $j$-th eigenvalue is of order $\left(\log^{p-1}(j+1)/j\right)^α$. For $ c_1 d \log^{-(p+1)}(d)\leq j\leq d$, the $j$-th eigenvalue is of order $j^{-α}$ up to a polylog factor. That is, the power-law exponent $α$ is inherited exactly from the input covariance, modified only by a logarithmic correction that depends on the monomial degree $p$. The proof combines a dyadic head-tail decomposition with Wick chaos expansions for higher-order monomials and random matrix concentration inequalities.
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PROBLEM
This paper explores the spectral structure of data covariance in neural networks through the random feature model. A central question is whether this spectral structure is preserved or destroyed when data passes through the basic building block of a neural network: a random line...
METHOD
Scaling laws for neural networks, in which the loss decays as a power-law in the number of parameters, data, and compute, depend fundamentally on the spectral structure of the data covariance, with power-law eigenvalue decay appearing ubiquitously in vision and language tasks. A...
RESULT
ScienceToStartup currently rates this 2.0/10 on the public viability pass. The proof combines a dyadic head-tail decomposition with Wick chaos expansions for higher-order monomials and random matrix concentration inequalities.
WHY NOW
Theoretical Analysis moved forward this cycle; last verified April 2026. Public score 2.0/10.
Abstract-backed public claims while anchored extraction refreshes.
This paper explores the spectral structure of data covariance in neural networks through the random feature model. A central question is whether this spectral structure is preserved or destroyed when data passes through the basic building block of a neural network: a random linear projection followed by a nonlinear activation.
Abstract-backed fallback claim; anchored extraction has not materialized a public claim row yet.
partial
Scaling laws for neural networks, in which the loss decays as a power-law in the number of parameters, data, and compute, depend fundamentally on the spectral structure of the data covariance, with power-law eigenvalue decay appearing ubiquitously in vision and language tasks. A central question is whether this spectral structure is preserved or destroyed when data passes through the basic building block of a neural network: a random linear projection followed by a nonlinear activation.
Abstract-backed fallback claim; anchored extraction has not materialized a public claim row yet.
partial
ScienceToStartup currently rates this 2.0/10 on the public viability pass. The proof combines a dyadic head-tail decomposition with Wick chaos expansions for higher-order monomials and random matrix concentration inequalities.
Abstract-backed fallback claim; anchored extraction has not materialized a public claim row yet.
partial
Theoretical Analysis moved forward this cycle; last verified April 2026. Public score 2.0/10.
Abstract-backed fallback claim; anchored extraction has not materialized a public claim row yet.
partial
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This paper explores the spectral structure of data covariance in neural networks through the random feature model.
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Theoretical Analysis
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