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Spectral Estimation with Free Decompression

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Spectral densities of extremely large or inaccessible 'impalpable' Hermitian matrices can be estimated from the spectral density of a small randomly sampled submatrix by evolving its Stieltjes transform through a quasilinear PDE in the…

desk verdict Free decompression is a clever numerical pipeline built on a known free-probability result, but the paper overclaims its scope: random coordinate submatrices of arbitrary deterministic matrices are not free compressions, and the diagonal-matrix counterexample breaks the central method. read the letter →

arxiv 2506.11994 v1 pith:FAM457VM submitted 2025-06-13 stat.ML cs.LGcs.NAmath.NA

Siavash Ameli , Chris van der Heide , Liam Hodgkinson , Michael W. Mahoney This is my paper
classification stat.MLcs.LGcs.NAmath.NA MSC 60B2046L54
keywords spectralestimationfreeprobabilityStieltjestransformR-transformdecompressionimpalpablematricesrandommatrixtheory
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Free decompression claims to estimate the empirical spectral density of a Hermitian matrix so large or so inaccessible that it can neither be formed nor applied to a vector — what the paper calls an impalpable matrix — using only the eigenvalues of a small randomly sampled submatrix. The engine is a free-probability identity: by the Nica-Speicher theorem, taking a submatrix compresses the R-transform of a matrix by a linear scaling of its argument, and the paper runs that operation in reverse. In Stieltjes-transform form the reversal becomes a first-order quasilinear PDE in the log of the matrix size, $\partial m/\partial t = -m + m^{-1}\partial m/\partial z$, solved by the method of characteristics to extrapolate the submatrix's spectral density to the full matrix's. Errors grow at most polynomially in the decompression ratio $n/n_s$ (Proposition 3), but only under the load-bearing assumption, stated rather than proved for arbitrary matrices, that the sampled submatrix is a free compression of the full matrix. Demonstrated on five classical random-matrix limits, a real graph Laplacian, and an empirical neural tangent kernel, the method would, if valid, open a route to log-determinants, traces of matrix functions, and other spectral invariants when no matrix-vector product is available.

What carries the argument

The engine is the Stieltjes transform $m(z) = n^{-1}\mathrm{tr}((A - zI)^{-1})$ together with Voiculescu's R-transform, the functional inverse relation that encodes the spectral density of a Hermitian matrix. The Nica-Speicher theorem supplies the scaling $R_{n_s}(z) = R(z\, n_s/n)$ for free compression by a submatrix; free decompression inverts it to $R_n(z) = R_{n_s}((n/n_s)z)$. In Stieltjes-transform variables this inversion is the quasilinear PDE $\partial m/\partial t = -m + m^{-1}\partial m/\partial z$, solved by the method of characteristics with $t(\tau) = \tau$, $z(\tau) = z_0 - m_0(z_0)^{-1}(e^{\tau} - 1)$, $m(\tau) = m_0(z_0)e^{-\tau}$. Because the characteristic curves of this PDE cross the real axis through the support of the spectral density, where the principal branch of the Stieltjes transform has a branch cut, the implementation needs a second analytic sheet (the 'secondary branch') built from a rational gluing function $G(z)$ chosen to cancel the jump. The initial density is smoothed in a Chebyshev or Jacobi basis, with the closed-form Stieltjes transform $\pi J(z)^{k+1}$ for the basis elements $U_k(x)\sqrt{1-x^2}$, so that the initial transform has no poles on the real axis. Proposition 3's polynomial error bound $\|\delta m_n\|_{L^2} \leq (n/n_s)^{\nu}\|\delta m_{n_s}\|_{L^2}$ is what licenses large decompression ratios.

What would settle it

Use a large deterministic Hermitian matrix whose structure is as far from 'free' as possible — for instance a diagonal matrix with entries drawn from a heavy-tailed distribution, where a random principal submatrix simply subsamples eigenvalues rather than compressing them freely. Free decompression from that submatrix will produce a predicted density that differs from the true spectrum far beyond the Proposition 3 polynomial bound; demonstrating that gap shows the freeness assumption, not the PDE machinery, is the operative constraint.

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Extended reading notes

Core claim

The central claim is that spectral information compresses in a computable way: by the Nica-Speicher theorem, for a free random matrix the R-transform of a top-left submatrix holding a fraction $\alpha$ of the rows and columns is $R_{\alpha n}(z) = R(\alpha z)$, a rule called free compression. The paper's contribution is to reverse this rule — free decompression — so that from the R-transform (equivalently the Stieltjes transform) of an $n_s \times n_s$ sampled submatrix one obtains $R_n(z) = R_{n_s}((n/n_s)z)$ for the full matrix. In Stieltjes-transform form the reversal is the quasilinear PDE $\partial m/\partial t = -m + m^{-1}\partial m/\partial z$ with $t = \log(n/n_s)$, whose explicit characteristic solution $m(t,z) = m_0(\varphi(t,z))e^{-t}$ maps the submatrix density to the full-matrix density. The paper asserts this evolution reproduces the empirical spectral density of the large matrix, proves that $L^2$ error grows at most polynomially in $n/n_s$ (Proposition 3), and demonstrates the procedure on five random-matrix ensembles with known limiting laws, on the Laplacian spectrum of a real social network, and on an empirical neural tangent kernel whose full spectrum is matched from an initial $2048 \times 2048$ submatrix.

Load-bearing premise

The load-bearing premise is that a randomly sampled principal submatrix of the matrix of interest is a free compression of it, meaning the matrix sequence is asymptotically free in the sense of free probability; this is asserted for deterministic matrices rather than proved, and if it fails the PDE extrapolation can be arbitrarily wrong.

Editorial extensions

If this is right

  • Log-determinants, traces of matrix functions, and other spectral invariants become computable for impalpable matrices, since the full spectral density is recovered before any invariant is evaluated.
  • The PDE continues in both directions along the size axis: a spectrum estimated from an $n \times n$ matrix can be decompressed further to predict matrices larger than the ones actually observed, not only smaller.
  • No matrix-vector product is ever needed; the only linear-algebra work is the eigendecomposition of the small sampled submatrix.
  • Accuracy for real data is governed by the quality of the smoothed density estimate and of the gluing-function continuation, which the paper's limitations section identifies as the delicate stages of the pipeline.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The freeness assumption is directly testable: applying free decompression to structured matrices such as diagonal, banded, or low-rank-plus-diagonal matrices — where a coordinate submatrix provably is not a free compression — should fail in a measurable way, and mapping where it fails would delimit the method's validity region.
  • Because decompression reconstructs the limiting free-probability law of the matrix sequence rather than the exact finite-$n$ empirical spectrum, its output for a single finite matrix is best read as the spectrum the matrix 'would have' if its structural correlations were negligible; users with strongly structured matrices should expect bias of exactly that kind.
  • The method offers a principled replacement for the empirical scaling-law extrapolation of log-determinants used in earlier work on impalpable matrices: what was previously an observed power-law trend is here derived from the PDE, so the two approaches could be merged to turn scaling-law fits into parameter estimates.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper introduces "free decompression," a method to estimate the empirical spectral density of a large Hermitian matrix from the spectral density of a smaller randomly sampled principal submatrix. Under an asymptotic freeness assumption for the submatrix selection, the authors show that the Stieltjes transform evolves according to a quasilinear PDE (Proposition 1) with an explicit characteristic solution (Proposition 2), and they state a polynomial error-growth bound (Proposition 3). Numerical experiments cover synthetic random matrix ensembles (Wigner, Marchenko-Pastur, Kesten-McKay, Wachter, Meixner), a Facebook Page-Page graph Laplacian, and an empirical neural tangent kernel. The paper claims applicability to "virtually any class of matrices," including arbitrary deterministic matrices accessible only through submatrices.

Significance. The derivation of the PDE from R-transform scaling is elegant, the characteristic solution is explicit, and the paper ships a Python package (freealg) that reproduces the experiments. The synthetic experiments are consistent with the theory for random matrix ensembles. However, the central assumption that a random coordinate submatrix of an arbitrary deterministic matrix is asymptotically free is unproven and demonstrably false for matrices with localized eigenvectors. This limits the significance to settings where eigenvector delocalization can be established; as written, the advertised scope is broader than what is supported. The error-growth result of Proposition 3 also needs a more careful proof.

major comments (2)
  1. [Section 3 (paragraph beginning "We emphasize that the matrix A itself may be arbitrary and deterministic")] The paper's key modeling assumption is that random coordinate submatrix selection makes an arbitrary deterministic Hermitian matrix asymptotically free from the coordinate projection (Sections 2 and 3). This is not proven and is false in general. For example, let A = diag(λ_1,...,λ_n) with a limiting empirical spectral measure μ. A uniformly random coordinate submatrix of size ns has eigenvalues that are a random sample from μ, so its ESD converges to μ, not to the free compression of μ with ratio ns/n. Running free decompression on this initial density yields the free compression of μ (e.g., a Marchenko-Pastur law with ratio λ·n/ns when μ is MP(λ)), which differs from μ. Thus the method is systematically wrong for such matrices even though the submatrix data are perfectly healthy. The abstract's claim "virtually any class of matrices" and the statement that A "may be arbitrary and deterministic" are therefore unsupported. The paper must either restrict the scope to matrices with sufficiently delocalized eigenvectors (or to random matrix ensembles where freeness holds) or provide a verifiable sufficient condition and prove it for the examples.
  2. [Appendix F, Proposition 3] The proof of Proposition 3 is not complete. Proposition F.1 assumes the perturbation δm satisfies the Bernstein inequality (F.1), i.e., that δm is band-limited on each horizontal slice; differences of Stieltjes transforms of probability measures do not generally satisfy this, so this is not a "mild assumption" as claimed. Moreover, the derived exponent ν = -2 + C_y(t) depends on t and y, and the text does not show that C_y(t) is bounded uniformly in t or as y → 0+. Without such a uniform bound, the conclusion that ∥δm(t)∥ grows at most polynomially in n/ns is not established. The statement of Proposition 3 should be revised to state precise hypotheses and the dependence of ν, or the proof needs to supply the missing uniform bounds.
minor comments (5)
  1. [Section 3] The phrase "method of sumbatrix selection" contains a typo; it should read "submatrix selection."
  2. [Figure 3 caption] The notation "n = 212" is ambiguous; it should be written as "n = 2^12" (and similarly for the other powers) to avoid confusion.
  3. [Appendix D.1] The word "triadiagonal" should be "tridiagonal".
  4. [Section 6 (Limitations)] The statement "no competing alternative exists for estimating the spectral density of impalpable matrices" is too strong and is not supported by a literature comparison; it should be softened or substantiated.
  5. [Table 2] The table header "ns Direct FD (ours)" is unclear; it should explicitly separate the submatrix size ns from the two processes being compared.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the free decompression estimate is a parameter-free PDE extrapolation built on an external free-probability theorem, and the paper's self-citations are background only.

full rationale

The derivation chain is: the submatrix Stieltjes transform is computed from the sampled submatrix spectrum; Theorem 1 (Nica–Speicher, credited to Nica & Speicher 1996 and Olver & Nadakuditi 2012) gives the free-compression scaling of the R-transform; Proposition 1 derives the PDE (1) from that scaling, Proposition 2 solves it by characteristics, and Proposition 3 bounds perturbation growth. All of this is external mathematics or calculus from it, and no step uses the target full spectrum to fit a parameter. The hyperparameters (Jacobi exponents, Padé degrees, Jackson damping, regularization) are fit to the submatrix density or chosen a priori; the full spectrum appears only as an external benchmark in the experiments. The key freeness assumption is stated openly as an assumption ('This corresponds to an assumption that the matrices are part of a sequence that is asymptotically free (Maïda, 2023)'), not derived from the authors' prior work; if it fails, the PDE is misspecified, which is a model-risk concern, not a circular derivation. Self-citations (e.g., Ameli et al. 2025 for FLODANCE and NTK ill-conditioning; Hodgkinson et al. 2023a for log-determinant scaling) are motivational and do not carry the load-bearing derivation. The Limitations section candidly acknowledges sensitivity to spectral-density estimation and gluing-function quality, and it does not claim independence from those inputs. Therefore no circular step can be exhibited: the paper's central claim reduces neither to its inputs by construction nor to a self-citation chain.

Assumptions & free parameters 6 free parameters · 5 assumptions · 2 invented entities

The central claim rests on the Nica-Speicher theorem combined with an asymptotic freeness assumption for submatrix selection. The free parameters are numerical hyperparameters for density and glue-function approximation, not fitted against the target full spectrum. The invented entities are mathematical tools for analytic continuation, not physical objects.

free parameters (6)
  • Jacobi polynomial shape parameters alpha, beta = 0.5 (default in Listing 1)
    Shape parameters for the Jacobi polynomial basis; chosen by hand, affect the initial density approximation and hence the decompression.
  • Polynomial degree K = 20 in Listing 1
    Truncation order of the spectral density expansion; larger K improves accuracy but risks Gibbs oscillations.
  • Regularization gamma = small, 0 in Listing 1 by default
    Tikhonov penalty on polynomial coefficients; tuned to stabilize the estimate.
  • Pade degrees p,q for glue function = p=0,q=1 in Listing 1
    Rational approximation of the Hilbert transform and secondary branch; the paper states the method is sensitive to this choice.
  • Perturbation delta = small positive, 0<delta<<1
    Imaginary offset in Algorithm 1 to regularize the characteristic solver; hand-chosen.
  • Support endpoints lambda_plus, lambda_minus = estimated from submatrix min/max eigenvalues with padding
    Endpoint estimates of the support interval; data-dependent and can bias the density if the submatrix is not representative.
assumptions (5)
  • standard math Nica-Speicher free compression theorem: for a free random matrix A with R-transform R(z), the R-transform of the top-left ns x ns submatrix is R(z ns/n).
    Invoked as the foundation of free decompression in Section 3.1. It is a known theorem from free probability, so not ad hoc, but its applicability requires the matrix and the sampling projection to be free.
  • domain assumption The sequence of matrices is asymptotically free, meaning a random principal submatrix of a deterministic Hermitian matrix behaves like a free compression of the full matrix.
    Stated in Section 2: 'This corresponds to an assumption that the matrices are part of a sequence that is asymptotically free.' This is the load-bearing assumption for extending the Nica-Speicher result to arbitrary deterministic matrices.
  • domain assumption The spectral density has compact support, is bounded in L-infinity, and is piecewise C^1.
    Used in Proposition 3 and Appendix C for Hilbert transform properties and stability bounds. Standard for the random matrix ensembles considered, but not true for all matrices (e.g., point masses, unbounded support).
  • standard math The Stieltjes transform of the initial density has no zeros on the domain Omega and admits an analytic continuation across the support.
    Required for Proposition 2 (m0(z) != 0) and for the secondary branch construction in Appendix C.2. The zero-free condition is not guaranteed in general.
  • domain assumption Hutchinson trace estimation variance is small enough to replace expectations by the mean of diagonal entries in the Schur complement heuristic.
    Used in Section 3.3 to derive the difference equation (4). Standard for large n but not rigorous for short widths.
invented entities (2)
  • Glue function G(z)
    purpose: Defines the secondary analytic branch of the Stieltjes transform by matching the real part of the Hilbert transform across the support.
    Introduced in Appendix C.2 as an additive correction; no external falsifiable content, it is a numerical construction.
  • Secondary branch m- of the Stieltjes transform
    purpose: Allows evaluation of m0 in the lower half-plane so characteristic curves can cross the real axis on the support.
    Mathematical continuation needed to implement the PDE solution; no independent evidence.

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Cite this review

Pith. "Pith review of Spectral Estimation with Free Decompression." pith.science (2026). https://pith.science/paper/FAM457VM

@misc{pith2026250611994,
  author       = {Pith},
  title        = {Pith review of: Spectral Estimation with Free Decompression},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FAM457VM}},
  note         = {Machine review of arXiv:2506.11994}
}
read the original abstract

Computing eigenvalues of very large matrices is a critical task in many machine learning applications, including the evaluation of log-determinants, the trace of matrix functions, and other important metrics. As datasets continue to grow in scale, the corresponding covariance and kernel matrices become increasingly large, often reaching magnitudes that make their direct formation impractical or impossible. Existing techniques typically rely on matrix-vector products, which can provide efficient approximations, if the matrix spectrum behaves well. However, in settings like distributed learning, or when the matrix is defined only indirectly, access to the full data set can be restricted to only very small sub-matrices of the original matrix. In these cases, the matrix of nominal interest is not even available as an implicit operator, meaning that even matrix-vector products may not be available. In such settings, the matrix is "impalpable," in the sense that we have access to only masked snapshots of it. We draw on principles from free probability theory to introduce a novel method of "free decompression" to estimate the spectrum of such matrices. Our method can be used to extrapolate from the empirical spectral densities of small submatrices to infer the eigenspectrum of extremely large (impalpable) matrices (that we cannot form or even evaluate with full matrix-vector products). We demonstrate the effectiveness of this approach through a series of examples, comparing its performance against known limiting distributions from random matrix theory in synthetic settings, as well as applying it to submatrices of real-world datasets, matching them with their full empirical eigenspectra.

Figures

Figures reproduced from arXiv: 2506.11994 by the authors.

Figure 1
Figure 1. (a) Estimation of the Marchenko–Pastur law with ratio [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Analytic continuation of the Stieltjes transform of a Marchenko–Pastur distribution (with ratio [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. (a) Empirical spectral densities (dashed) of the symmetrically normalized Laplacian matrix of [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

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Works this paper leans on

72 extracted references · 67 canonical work pages

  1. [1]

    P., Pennington, J., Johnson, M

    Adams, R. P., Pennington, J., Johnson, M. J., Smith, J., Ovadia, Y., Patton, B., & Saunderson, J. (2018). Estimating the spectral density of large implicit matrices. arXiv preprint arXiv:1802.03451

    arXiv 2018

  2. [2]

    & Vitter, Jeffrey, S

    Aggarwal, A. & Vitter, Jeffrey, S. (1988). The input/output complexity of sorting and related problems. Communications of the ACM , 31(9), 1116--1127

    work page 1988

  3. [3]

    Ameli, S., van der Heide, C., Hodgkinson, L., Roosta, F., & Mahoney, M. W. (2025). Determinant estimation under memory constraints and neural scaling laws. arXiv preprint arXiv:2503.04424

    work page Pith review arXiv 2025

  4. [4]

    Anshelevich, M. (2007). Free Meixner states. Communications in Mathematical Physics , 276(3), 863--899

    work page 2007

  5. [5]

    Anshelevich, M. (2008). Orthogonal polynomials with a resolvent-type generating function. Transactions of the American Mathematical Society , 360(8), 4125--4143

    work page 2008

  6. [6]

    Bekas, C., Kokiopoulou, E., & Saad, Y. (2007). An estimator for the diagonal of a matrix. Applied Numerical Mathematics , 57, 1214--1229

    work page 2007

  7. [7]

    Bhattacharjee, R., Jayaram, R., Musco, C., Musco, C., & Ray, A. (2025). Improved spectral density estimation via explicit and implicit deflation. In Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) (pp.\ 2693--2754).: SIAM

    work page 2025

  8. [8]

    T., Liang, T., & Zhou, H

    Cai, T. T., Liang, T., & Zhou, H. H. (2015). Law of log determinant of sample covariance matrix and optimal estimation of differential entropy for high-dimensional G aussian distributions. Journal of Multivariate Analysis , 137, 161--172

    work page 2015

Show all 72 references
  1. [9]

    & Utard, G

    Caron, E. & Utard, G. (2002). Parallel out-of-core matrix inversion. In Proceedings 16th International Parallel and Distributed Processing Symposium (pp.\ 6--pp).: IEEE

    2002

  2. [10]

    Chen, T., Trogdon, T., & Ubaru, S. (2021). Analysis of stochastic Lanczos quadrature for spectrum approximation. In M. Meila & T. Zhang (Eds.), Proceedings of the 38th International Conference on Machine Learning , volume 139 of Proceedings of Machine Learning Research (pp.\ 1...

    2021

  3. [11]

    Chizat, L., Oyallon, E., & Bach, F. (2019). On lazy training in differentiable programming. In H. Wallach, H. Larochelle, A. Beygelzimer, F. d Alch\' e -Buc, E. Fox, & R. Garnett (Eds.), Advances in Neural Information Processing Systems , volume 32: Curran Associates, Inc

    2019

  4. [12]

    & Liao, Z

    Couillet, R. & Liao, Z. (2022). Random Matrix Methods for Machine Learning . Cambridge University Press. https://zhenyu-liao.github.io/book/

    2022

  5. [13]

    & Mahoney, M

    Derezi \'n ski, M. & Mahoney, M. W. (2021). Determinantal point processes in randomized numerical linear algebra. Notices of the AMS , 68(1), 34--45

    2021

  6. [14]

    & Mahoney, M

    Drineas, P. & Mahoney, M. W. (2016). RandNLA : Randomized numerical linear algebra. Communications of the ACM , 59, 80--90

    2016

  7. [15]

    W., & Muthukrishnan, S

    Drineas, P., Mahoney, M. W., & Muthukrishnan, S. (2008). Relative-error CUR matrix decompositions. SIAM Journal on Matrix Analysis and Applications , 30, 844--881

    2008

  8. [16]

    & Edelman, A

    Dubbs, A. & Edelman, A. (2015). Infinite random matrix theory, tridiagonal bordered Toeplitz matrices, and the moment problem. Linear Algebra and its Applications , 467, 188--201

    2015

  9. [17]

    & Schweitzer, M

    Frommer, A. & Schweitzer, M. (2016). Error bounds and estimates for Krylov subspace approximations of Stieltjes matrix functions. BIT Numerical Mathematics , 56(3), 865--892

    2016

  10. [18]

    Q., Bindel, D., & Wilson, A

    Gardner, J., Pleiss, G., Weinberger, K. Q., Bindel, D., & Wilson, A. G. (2018). GPyTorch: Blackbox matrix-matrix Gaussian process inference with GPU acceleration . Advances in neural information processing systems , 31

    2018

  11. [19]

    W., & Keutzer, K

    Gholami, A., Yao, Z., Kim, S., Hooper, C., Mahoney, M. W., & Keutzer, K. (2024). Ai and memory wall. IEEE Micro , 44(3), 33–39

    2024

  12. [20]

    W., & Prabhat (2016)

    Gittens, A., Devarakonda, A., Racah, E., Ringenburg, M., Gerhardt, L., Kottalam, J., Liu, J., Maschhoff, K., Canon, S., Chhugani, J., Sharma, P., Yang, J., Demmel, J., Harrell, J., Krishnamurthy, V., Mahoney, M. W., & Prabhat (2016). Matrix factorizations at scale: A compariso...

    2016

  13. [21]

    & Mahoney, M

    Gittens, A. & Mahoney, M. W. (2016). Revisiting the N ystr\" o m method for improved large-scale machine learning. Journal of Machine Learning Research , 17(117), 1--65

    2016

  14. [22]

    Golub, G. H. & Meurant, G. (1997). Matrices, moments and quadrature II ; how to compute the norm of the error in iterative methods. BIT , 37(3), 687–705

    1997

  15. [23]

    Golub, G. H. & Meurant, G. (2010). Matrices, Moments and Quadrature with Applications . Princeton University Press

    2010

  16. [24]

    Hamilton, R. S. (1982). The inverse function theorem of Nash and Moser . Bulletin (New Series) of the American Mathematical Society , 7(1), 65 -- 222

    1982

  17. [25]

    He, K., Zhang, X., Ren, S., & Sun, J. (2016). Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition (pp.\ 770--778)

    2016

  18. [26]

    Higham, N. (2022). Seven Sins of Numerical Linear Algebra . https://nhigham.com/2022/10/11/seven-sins-of-numerical-linear-algebra/. Accessed: 2025-01-29

    2022

  19. [27]

    Hodgkinson, L., van der Heide, C., Roosta, F., & Mahoney, M. W. (2023a). Monotonicity and double descent in uncertainty estimation with G aussian processes. In Proceedings of the 40th International Conference on Machine Learning (pp.\ 13085--13117)

    2023

  20. [28]

    Hodgkinson, L., van der Heide, C., Salomone, R., Roosta, F., & Mahoney, M. W. (2023b). The interpolating information criterion for overparameterized models. arXiv preprint arXiv:2307.07785v1

    2023

  21. [29]

    Hodgkinson, L., van der Heide, C., Salomone, R., Roosta, F., & Mahoney, M. W. (2023c). A PAC-Bayesian perspective on the interpolating information criterion. In Advances in Neural Information Processing Systems , Mathematics of Modern Machine Learning Workshop

    2023

  22. [30]

    & Landon, B

    Huang, J. & Landon, B. (2020). Spectral statistics of sparse Erd o s-R\' e nyi graph Laplacians . Annales de l'Institut Henri Poincar \'e , Probabilit \'e s et Statistiques , 56(1), 120--154

    2020

  23. [31]

    Immer, A., Korzepa, M., & Bauer, M. (2021). Improving predictions of Bayesian neural nets via local linearization. In A. Banerjee & K. Fukumizu (Eds.), Proceedings of The 24th International Conference on Artificial Intelligence and Statistics , volume 130 of Proceedings of Mac...

    2021

  24. [32]

    Immer, A., Van Der Ouderaa, T. F. A., Van Der Wilk, M., Ratsch, G., & Sch\" o lkopf, B. (2023). Stochastic marginal likelihood gradients using neural tangent kernels. In A. Krause, E. Brunskill, K. Cho, B. Engelhardt, S. Sabato, & J. Scarlett (Eds.), Proceedings of the 40th In...

    2023

  25. [33]

    Jackson, D. (1912). On approximation by trigonometric sums and polynomials. Transactions of the American Mathematical Society , 13(4), 491--515

    1912

  26. [34]

    Jacot, A., Gabriel, F., & Hongler, C. (2018). Neural tangent kernel: Convergence and generalization in neural networks. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, & R. Garnett (Eds.), Advances in Neural Information Processing Systems , volume 31: Cur...

    2018

  27. [35]

    Kesten, H. (1959). Symmetric random walks on groups. Transactions of the American Mathematical Society , 92(2), 336--354

    1959

  28. [36]

    Kim, S., Park, S., Kim, K.-S., & Yang, E. (2023). Scale-invariant Bayesian neural networks with connectivity tangent kernel. In The Eleventh International Conference on Learning Representations

    2023

  29. [37]

    King, F. W. (2009). Hilbert Transforms . Encyclopedia of Mathematics and its Applications. Cambridge University Press

    2009

  30. [38]

    Krizhevsky, A. (2009). Learning multiple layers of features from tiny images . Technical Report TR-2009, University of Toronto

    2009

  31. [39]

    Lenczewski, R. (2015). Limit distributions of Gaussian block ensembles. Acta Physica Polonica B , 46(9), 1833--1850

    2015

  32. [40]

    & Krevl, A

    Leskovec, J. & Krevl, A. (2014). SNAP Datasets : Stanford large network dataset collection. http://snap.stanford.edu/data

    2014

  33. [41]

    Longoria, I. S. A. & and, J. A. M. (2023). Freely independent coin tosses, standard Young tableaux, and the Kesten--McKay law. The American Mathematical Monthly , 130(1), 35--48

    2023

  34. [42]

    Lowe, D., Kim, M., & Bondesan, R. (2025). Assessing quantum advantage for G aussian process regression. arXiv preprint arXiv:2505.22502

    2025 arXiv

  35. [43]

    Mahoney, M. W. (2016). Lecture notes on spectral graph methods. arXiv preprint arXiv:1608.04845

    2016 arXiv

  36. [44]

    Ma \" da, M. (2023). Strong convergence of the spectrum of random permutations and almost-Ramanujan graphs . S \'e minaire Bourbaki , (pp.\ 75e)

    2023

  37. [45]

    Mar c enko, V. A. & Pastur, L. A. (1967). Distribution of eigenvalues for some sets of random matrices. Mathematics of the USSR-Sbornik , 1(4), 457

    1967

  38. [46]

    McKay, B. D. (1981). The expected eigenvalue distribution of a large regular graph. Linear Algebra and its Applications , 40, 203--216

    1981

  39. [47]

    Meixner, J. (1934). Orthogonale polynomsysteme mit einer besonderen gestalt der erzeugenden funktion. Journal of The London Mathematical Society-second Series , (pp.\ 6--13)

    1934

  40. [48]

    Meurant, G. (2006). The Lanczos and Conjugate Gradient Algorithms: From Theory to Finite Precision Computations (Software, Environments, and Tools) . USA: Society for Industrial and Applied Mathematics

    2006

  41. [49]

    Mezzadri, F. (2007). How to generate random matrices from the classical compact groups. Notices of the American Mathematical Society , 54(5), 592--604

    2007

  42. [50]

    Mu, X., Zhou, H.-X., Chen, K., & Hong, W. (2014). Higher order method of moments with a parallel out-of-core LU solver on GPU/CPU platform. IEEE Transactions on Antennas and Propagation , 62(11), 5634--5646

    2014

  43. [51]

    W., Erichson, N

    Murray, R., Demmel, J., Mahoney, M. W., Erichson, N. B., Melnichenko, M., Malik, O. A., Grigori, L., Luszczek, P., Derezi \'n ski, M., Lopes, M. E., Liang, T., Luo, H., & Dongarra, J. (2023). Randomized Numerical Linear Algebra : A perspective on the field with an eye to softw...

    2023 arXiv

  44. [52]

    Nguyen, H. H. & Vu, V. (2014). Random matrices: Law of the determinant. The Annals of Probability , (pp.\ 146--167)

    2014

  45. [53]

    Nica, A. (1993). Asymptotically free families of random unitaries in symmetric groups. Pacific Journal of Mathematics , 157(2)

    1993

  46. [54]

    & Speicher, R

    Nica, A. & Speicher, R. (1996). On the multiplication of free n -tuples of noncommutative random variables. American Journal of Mathematics , 118(4), 799--832

    1996

  47. [55]

    Novak, R., Sohl-Dickstein, J., & Schoenholz, S. S. (2022). Fast finite width neural tangent kernel. In International Conference on Machine Learning (pp.\ 17018--17044).: PMLR

    2022

  48. [56]

    & Nadakuditi, R

    Olver, S. & Nadakuditi, R. R. (2012). Numerical computation of convolutions in free probability theory. arXiv preprint arXiv:1203.1958

    2012 arXiv

  49. [57]

    Potapczynski, A., Finzi, M., Pleiss, G., & Wilson, A. G. (2023). Cola: Exploiting compositional structure for automatic and efficient numerical linear algebra. In A. Oh, T. Naumann, A. Globerson, K. Saenko, M. Hardt, & S. Levine (Eds.), Advances in Neural Information Processin...

    2023

  50. [58]

    & Toledo, S

    Rabani, E. & Toledo, S. (2001). Out-of-core SVD and QR decompositions. In PPSC

    2001

  51. [59]

    & Ascher, U

    Roosta-Khorasani, F. & Ascher, U. (2015). Improved bounds on sample size for implicit matrix trace estimators. Foundations of Computational Mathematics , 15(5), 1187--1212

    2015

  52. [60]

    Rozemberczki, B., Allen, C., & Sarkar, R. (2021). Multi-scale attributed node embedding. Journal of Complex Networks , 9(2), cnab014

    2021

  53. [61]

    K., Alexanderian, A., & Ipsen, I

    Saibaba, A. K., Alexanderian, A., & Ipsen, I. C. (2017). Randomized matrix-free trace and log-determinant estimators. Numerische Mathematik , 137(2), 353--395

    2017

  54. [62]

    & Yoshida, H

    Saitoh, N. & Yoshida, H. (2001). The infinite divisibility and orthogonal polynomials with a constant recursion formula in free probability theory. Probab. Math. Statist. , 21, 159--170

    2001

  55. [63]

    Shen, J., Tang, T., & Wang, L.-L. (2011). Spectral Methods: Algorithms, Analysis and Applications . Springer Publishing Company, Incorporated, 1st edition

    2011

  56. [64]

    Tao, T. (2017). Inverting the Schur complement, and large-dimensional Gelfand-Tsetlin patterns

    2017

  57. [65]

    Trefethen, L. N. (2023). Numerical analytic continuation. Japan Journal of Industrial and Applied Mathematics , 40(3), 1587--1636

    2023

  58. [66]

    Ubaru, S., Chen, J., & Saad, Y. (2017). Fast estimation of tr(f(a)) via stochastic Lanczos quadrature. SIAM Journal on Matrix Analysis and Applications , 38(4), 1075--1099

    2017

  59. [67]

    & Saad, Y

    Ubaru, S. & Saad, Y. (2017). Applications of trace estimation techniques. In International Conference on High Performance Computing in Science and Engineering (pp.\ 19--33).: Springer

    2017

  60. [68]

    R., shan Shiu, D., & Bernacchia, A

    Vakili, S., Bromberg, M., Garcia, J. R., shan Shiu, D., & Bernacchia, A. (2022). Uniform generalization bounds for overparameterized neural networks

    2022

  61. [69]

    Voiculescu, D. (1991). Limit laws for Random matrices and free products . Inventiones mathematicae , 104(1), 201--220

    1991

  62. [70]

    Wachter, K. W. (1978). The strong limits of random matrix spectra for sample matrices of independent elements. The Annals of Probability , 6(1), 1--18

    1978

  63. [71]

    Wigner, E. P. (1955). Characteristic vectors of bordered matrices with infinite dimensions. Annals of Mathematics , 62(3), 548--564

    1955

  64. [72]

    Wilson, J., van der Heide, C., Hodgkinson, L., & Roosta, F. (2025). Uncertainty quantification with the empirical neural tangent kernel. arXiv preprint arXiv:2502.02870

    2025

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.