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arxiv: 2605.15240 · v1 · pith:63Y3ZC4Xnew · submitted 2026-05-14 · 📊 stat.ML · cs.LG

On Kernel Eigen-alignments of KRR: Reconstruction and Generalization

Yang Liu , Ernest Fokoue , Richard Lange , Daniel Krutz This is my paper

Pith reviewed 2026-05-19 16:20 UTC · model grok-4.3

classification 📊 stat.ML cs.LG
keywords kernel ridge regressioneigen-alignmentgeneralization boundseigenvalue estimationkernel methodsmatrix perturbation
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The pith

In kernel ridge regression, generalization performance depends on the alignment of kernel eigenvectors with the target vectors rather than just reconstruction accuracy.

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

This paper connects generalization in kernel methods to how well the eigenvectors and eigenvalues of the kernel matrix can be estimated from finite data. It shows that the prediction is a weighted sum of these eigenvectors, so perturbations in the kernel lead to errors bounded by eigenvalue stability. For high-rank kernels, low reconstruction error is easy to achieve and thus weakly predictive of generalization. Strong performance instead requires better eigenvector alignment with targets, larger eigenvalue magnitudes, or larger gaps between consecutive eigenvalues.

Core claim

The paper establishes that a generalization bound for kernel ridge regression can be derived from the estimation stability of eigenvalues and eigenvectors of the kernel matrix. Since predictions are weighted sums of eigenvectors, the error from kernel perturbations can be analyzed to show that strong generalization requires increasing eigenvector alignment, eigenvalue magnitude, or gaps between consecutive eigenvalues.

What carries the argument

eigen-alignments between the kernel matrix and learning targets, which determine how much the kernel perturbations affect the weighted eigenvector sums in predictions

If this is right

  • Low reconstruction error alone does not guarantee good generalization for high-rank kernels.
  • Improving eigenvector alignment enhances the generalization bound.
  • Larger gaps between eigenvalues increase stability against perturbations.
  • Increasing eigenvalue magnitudes supports better generalization.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This perspective suggests that kernel design should optimize for alignment properties in addition to other criteria.
  • Similar eigen-alignment analysis could extend to other kernel-based methods beyond ridge regression.
  • Finite-sample bounds like these may help explain why certain kernels work well on specific datasets but not others.

Load-bearing premise

The analysis assumes that the generalization error arises primarily from using a suboptimal finite training set in a finite-sample setting.

What would settle it

An experiment showing that generalization error remains high even when eigenvector alignment is strong and eigenvalues have large magnitudes and gaps would falsify the bound.

Figures

Figures reproduced from arXiv: 2605.15240 by Daniel Krutz, Ernest Fokoue, Richard Lange, Yang Liu.

Figure 1
Figure 1. Figure 1: Learning targets of the task affect the generalization performance. Learning targets aligned with top eigen [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Random data reconstruction error. We observe the following: 1) high reconstruction errors only appear when [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Learning targets of the task affect the generalization performance. Learning targets aligned with top eigen [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: With MNIST images data (labeled 4 and 9), we construct targets with top and trailing eigenvectors. On [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Training error and testing error comparison for different kernel types and eigenvectors as the learning targets. [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Left: Generalization error increases as the number of eigenvectors that the learning target aligned with [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Generalization error when treating each eigenvector as an individual synthetic learning target. [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
read the original abstract

This paper investigates the critical role of eigenalignments between the kernel matrix and learning targets in achieving robust generalization in learning problems. We establish a direct connection between generalization performance in kernel methods and the estimation of eigenvectors and eigenvalues of matrices, offering a more intuitive understanding compared to prior work with minimal assumptions. We also show that, since the prediction task in KRR is essentially the weighted sum of eigenvectors/singular vectors, by analyzing how much error can be caused by perturbations to the kernel matrix, we can then derive a bound on this generalization error using the estimation stability of matrix eigenvalues and eigenvectors. Compared with previous work, our analysis concentrates on finite-sample settings and on the generalization error arising from having a suboptimal finite training set. Our findings reveal that in kernel methods, as long as the kernel is of high rank, the near-zero reconstruction error can be trivially obtained, implying that the reconstruction error will have limited predictive power for generalization. Finally, we establish a generalization bound from an eigenvalues/eigenvectors estimation perspective, showing that strong generalization requires increasing eigenvector alignment, eigenvalue magnitude, or gaps between consecutive eigenvalues.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper analyzes kernel ridge regression (KRR) by linking generalization performance to eigen-alignments between the kernel matrix and learning targets. It derives a generalization bound by modeling the finite training set as inducing a perturbation on the kernel matrix, then bounding the error in the KRR predictor (expressed as a weighted sum of eigenvectors) via estimation stability of eigenvalues and eigenvectors. Key claims include that near-zero reconstruction error is trivial for high-rank kernels (limiting its predictive power for generalization) and that strong generalization requires increasing eigenvector alignment, eigenvalue magnitude, or gaps between consecutive eigenvalues, all under minimal assumptions in finite-sample settings.

Significance. If the perturbation analysis can be made rigorous with explicit conditions, the work would offer an intuitive finite-sample perspective on generalization in kernel methods that complements existing stability and Rademacher-based bounds. It could inform kernel selection or regularization by emphasizing eigenstructure, though the current lack of explicit perturbation form limits immediate applicability.

major comments (1)
  1. [perturbation analysis and generalization bound derivation] The central derivation of the generalization bound (described in the abstract as arising from perturbation stability of the kernel matrix and detailed in the main analysis) applies eigenvector perturbation results such as Davis-Kahan bounds without specifying the exact perturbation model (additive noise on K, sampling error from the integral operator, or multiplicative distortion) or verifying that eigenvalue gap conditions hold uniformly for the kernels under consideration. This is load-bearing for the claim that strong generalization requires larger alignment, larger eigenvalues, or larger gaps, as the bound does not necessarily follow from the finite-sample premise without these elements.
minor comments (1)
  1. [abstract and findings paragraph] The abstract states that 'near-zero reconstruction error can be trivially obtained' for high-rank kernels; clarify whether this holds only asymptotically or under specific finite-sample conditions, and add a brief remark on how this interacts with the proposed eigen-alignment requirements.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. We address the major comment on the perturbation analysis and generalization bound derivation below, and we plan to revise the paper to improve clarity on these points.

read point-by-point responses

  1. Referee: The central derivation of the generalization bound (described in the abstract as arising from perturbation stability of the kernel matrix and detailed in the main analysis) applies eigenvector perturbation results such as Davis-Kahan bounds without specifying the exact perturbation model (additive noise on K, sampling error from the integral operator, or multiplicative distortion) or verifying that eigenvalue gap conditions hold uniformly for the kernels under consideration. This is load-bearing for the claim that strong generalization requires larger alignment, larger eigenvalues, or larger gaps, as the bound does not necessarily follow from the finite-sample premise without these elements.

    Authors: We appreciate the referee highlighting the need for greater precision here. The perturbation we analyze is the additive difference between the empirical kernel matrix (computed from the finite training set) and the population kernel matrix induced by the integral operator; this is the natural sampling perturbation arising in the finite-sample setting. We will revise the manuscript to state this model explicitly, including a brief reference to standard concentration results for empirical kernel matrices. For the eigenvalue gap conditions, the Davis-Kahan bounds we invoke are applied conditionally on a positive gap between relevant eigenvalues, which is a standard technical assumption in this literature. We agree that discussing the uniformity of this condition strengthens the presentation. In revision we will add a short discussion noting that the bound holds whenever the gap condition is met and that regularization or kernel choice can be used to ensure suitable gaps for many common kernels (e.g., Gaussian). These changes make the assumptions transparent and support the claim that strong generalization requires sufficient alignment, eigenvalue magnitude, or gaps, without altering the core finite-sample perspective of the work. revision: yes

Circularity Check

0 steps flagged

Derivation applies perturbation analysis to KRR predictor without reducing to self-definition or fitted inputs

full rationale

The paper derives its generalization bound by expressing the KRR predictor as a weighted sum of eigenvectors and analyzing error from finite-sample perturbations to the kernel matrix, then invoking estimation stability results for eigenvalues and eigenvectors. This produces an explicit dependence of the bound on eigenvector alignment, eigenvalue magnitudes, and gaps, which is a direct consequence of the perturbation expansion rather than a tautological re-expression of the inputs. The separate observation that reconstruction error is near-zero for high-rank kernels is used only to downplay its predictive value for generalization and does not feed back into the bound derivation. No self-citations, ansatzes, or fitted parameters are shown to be load-bearing; the quantities in the bound are defined independently of the final generalization statement itself. The analysis therefore remains self-contained under the finite-sample premise stated in the abstract.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract alone: relies on standard matrix perturbation results for eigenvectors and eigenvalues; no free parameters, invented entities, or ad-hoc axioms explicitly named.

axioms (1)
  • standard math Standard results on stability of matrix eigenvalues and eigenvectors under perturbations hold in the finite-sample kernel setting.
    Invoked to derive the generalization bound from kernel matrix perturbations.

pith-pipeline@v0.9.0 · 5724 in / 1122 out tokens · 36126 ms · 2026-05-19T16:20:04.582258+00:00 · methodology

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Reference graph

Works this paper leans on

300 extracted references · 300 canonical work pages

  1. [1]

    Functional Analysis

    Rudin, Walter. Functional Analysis

    work page

  2. [2]

    On regularization algorithms in learning theory

    Bauer, Frank and Pereverzev, Sergei and Rosasco, Lorenzo. On regularization algorithms in learning theory. Journal of complexity

    work page

  3. [3]

    Approximation theory and harmonic analysis on spheres and balls

    Dai, Feng and Xu, Yuan. Approximation theory and harmonic analysis on spheres and balls

    work page

  4. [4]

    Aspects of Harmonic Analysis and Representation Theory

    Gallier, Jean and Quaintance, Jocelyn. Aspects of Harmonic Analysis and Representation Theory

    work page

  5. [5]

    Replica Method for the Machine Learning Theorist

    Blake Bordelon, Haozhe Shan, Abdul Canatar, Boaz Barak, Cengiz Pehlevan. Replica Method for the Machine Learning Theorist

    work page

  6. [6]

    Categorization based on similarity and features: the reproducing kernel Banach space (RKBS) approach

    Zhang, Jun and Zhang, Haizhang. Categorization based on similarity and features: the reproducing kernel Banach space (RKBS) approach. New Handbook of Mathematical Psychology

    work page

  7. [7]

    Fluctuations of Spiked Random Matrix Models and Failure Diagnosis in Sensor Networks

    Couillet, Romain and Hachem, Walid. Fluctuations of Spiked Random Matrix Models and Failure Diagnosis in Sensor Networks. IEEE Transactions on Information Theory

    work page

  8. [8]

    A Experimental details

    Datasets, Downstream. A Experimental details

    work page

  9. [9]

    Incremental kernel principal component analysis

    Chin, Tat-Jun and Suter, David. Incremental kernel principal component analysis. IEEE transactions on image processing: a publication of the IEEE Signal Processing Society

    work page

  10. [10]

    Scattered Data Approximation

    Wendland, Holger. Scattered Data Approximation

    work page

  11. [11]

    A support vector machine with a hybrid kernel and minimal vapnik-chervonenkis dimension

    Tan, Ying and Wang, Jun. A support vector machine with a hybrid kernel and minimal vapnik-chervonenkis dimension. IEEE transactions on knowledge and data engineering

    work page

  12. [12]

    Reproducing kernel Hilbert spaces on manifolds: Sobolev and Diffusion spaces

    De Vito, Ernesto and Mücke, Nicole and Rosasco, Lorenzo. Reproducing kernel Hilbert spaces on manifolds: Sobolev and Diffusion spaces. arXiv [math.FA]

    work page

  13. [13]

    Reproducing kernels of generalized Sobolev spaces via a Green function approach with distributional operators

    Fasshauer, Gregory E and Ye, Qi. Reproducing kernels of generalized Sobolev spaces via a Green function approach with distributional operators. Numerische mathematik

    work page

  14. [14]

    Adaptive kernel methods using the balancing principle

    De Vito, E and Pereverzyev, S and Rosasco, L. Adaptive kernel methods using the balancing principle. Foundations of computational mathematics

    work page

  15. [15]

    A Mathematical Introduction to Compressive Sensing

    Foucart, Simon and Rauhut, Holger. A Mathematical Introduction to Compressive Sensing

    work page

  16. [16]

    Matrix Perturbation Theory

    Stewart, G W and Sun, Ji-Guang. Matrix Perturbation Theory

    work page

  17. [17]

    Unshuffling data for improved generalization in visual question answering

    Teney, Damien and Abbasnejad, Ehsan and van den Hengel, Anton. Unshuffling data for improved generalization in visual question answering. 2021 IEEE/CVF International Conference on Computer Vision (ICCV)

    work page 2021

  18. [18]

    Dimension lower bounds for linear approaches to function approximation

    Hsu, Daniel J. Dimension lower bounds for linear approaches to function approximation

    work page

  19. [19]

    Measure Theory

    Cohn, Donald L. Measure Theory

    work page

  20. [20]

    for any p ∈ [1, ∞), n

    b‖, ‖a and Max, x :=. for any p ∈ [1, ∞), n

    work page

  21. [21]

    Algebraic geometry and statistical learning theory

    Watanabe, Sumio. Algebraic geometry and statistical learning theory

    work page

  22. [22]

    Measuring and testing homogeneity of distributions by characteristic distance

    Li, Xu and Hu, Wenjuan and Zhang, Baoxue. Measuring and testing homogeneity of distributions by characteristic distance. Statistical papers (Berlin, Germany)

    work page

  23. [23]

    Umbral nature of the Poisson random variables

    Di Nardo, E and Senato, D. Umbral nature of the Poisson random variables. arXiv [math.PR]

    work page

  24. [24]

    On the notion of neighborhood system

    Mishachev, Nikolai and Shmyrin, Anatoly. On the notion of neighborhood system. 2019 1st International Conference on Control Systems, Mathematical Modelling, Automation and Energy Efficiency (SUMMA)

    work page 2019

  25. [25]

    Functional Analysis for Probability and Stochastic Processes

    Bobrowski, A. Functional Analysis for Probability and Stochastic Processes

    work page

  26. [26]

    Stability and Generalization

    Bousquet, O and Elisseeff, A. Stability and Generalization. Journal of machine learning research: JMLR

    work page

  27. [27]

    Capacity of reproducing kernel spaces in learning theory

    Zhou, Ding-Xuan. Capacity of reproducing kernel spaces in learning theory. IEEE transactions on information theory

    work page

  28. [28]

    Approximation in learning theory

    Temlyakov, V N. Approximation in learning theory. Constructive approximation

    work page

  29. [29]

    Sobolev Norm Learning Rates for Regularized Least-Squares Algorithms

    Fischer, Simon and Steinwart, Ingo. Sobolev Norm Learning Rates for Regularized Least-Squares Algorithms. Journal of Machine Learning Research

    work page

  30. [30]

    Universal algorithms for learning theory part I : Piecewise constant functions

    Binev, P and Cohen, A and Dahmen, W and DeVore, R and Temlyakov, V. Universal algorithms for learning theory part I : Piecewise constant functions. Journal of machine learning research: JMLR

    work page

  31. [31]

    An introduction to Sobolev spaces

    Piskin, Erhan and Okutmustur, Baver. An introduction to Sobolev spaces

    work page

  32. [32]

    Entropy-based subsampling methods for big data

    Sui, Qun and Ghosh, Sujit K. Entropy-based subsampling methods for big data. Journal of statistical theory and practice

    work page

  33. [33]

    A theory of usable information under computational constraints

    Xu, Yilun and Zhao, Shengjia and Song, Jiaming and Stewart, Russell and Ermon, Stefano. A theory of usable information under computational constraints. arXiv [cs.LG]

    work page

  34. [34]

    Uniform approximation of functions with random bases

    Rahimi, Ali and Recht, Benjamin. Uniform approximation of functions with random bases. 2008 46th Annual Allerton Conference on Communication, Control, and Computing

    work page 2008

  35. [35]

    Tight frame expansions of multiscale reproducing kernels in Sobolev spaces

    Opfer, Roland. Tight frame expansions of multiscale reproducing kernels in Sobolev spaces. Applied and computational harmonic analysis

    work page

  36. [36]

    A theory of representation learning gives a deep generalisation of kernel methods

    Yang, Adam X and Robeyns, Maxime and Milsom, Edward and Anson, Ben and Schoots, Nandi and Aitchison, Laurence. A theory of representation learning gives a deep generalisation of kernel methods. Proceedings of the 40th International Conference on Machine Learning

    work page

  37. [37]

    Low rank spectral network alignment

    Nassar, Huda and Veldt, Nate and Mohammadi, Shahin and Grama, Ananth and Gleich, David F. Low rank spectral network alignment. Proceedings of the 2018 World Wide Web Conference on World Wide Web - WWW '18

    work page 2018

  38. [38]

    Learning the Kernel Function via Regularization

    Micchelli, Charles A and Pontil, Massimiliano. Learning the Kernel Function via Regularization. Journal of Machine Learning Research

    work page

  39. [39]

    HoMM: Higher-order Moment Matching for unsupervised domain adaptation

    Chen, Chao and Fu, Zhihang and Chen, Zhihong and Jin, Sheng and Cheng, Zhaowei and Jin, Xinyu and Hua, Xian-Sheng. HoMM: Higher-order Moment Matching for unsupervised domain adaptation. Proceedings of the ... AAAI Conference on Artificial Intelligence. AAAI Conference on Artificial Intelligence

    work page

  40. [40]

    Foundations of Machine Learning

    Mohri, Mehryar and Rostamizadeh, Afshin and Talwalkar, Ameet. Foundations of Machine Learning

    work page

  41. [41]

    Boosting a weak learning algorithm by majority

    Freund, Y. Boosting a weak learning algorithm by majority. Information and computation

    work page

  42. [42]

    Multivariate Function and Operator Estimation, Based on Smoothing Splines and Reproducing Kernels

    Wahba, Grace. Multivariate Function and Operator Estimation, Based on Smoothing Splines and Reproducing Kernels. Nonlinear Modeling and Forecasting

    work page

  43. [43]

    Smoothing Splines: Methods and Applications

    Wang, Yuedong. Smoothing Splines: Methods and Applications

    work page

  44. [44]

    Local Rademacher Complexity Bounds based on Covering Numbers

    Lei, Yunwen and Ding, Lixin and Bi, Yingzhou. Local Rademacher Complexity Bounds based on Covering Numbers. arXiv [cs.AI]

    work page

  45. [45]

    Concentration estimates for learning with ℓ1-regularizer and data dependent hypothesis spaces

    Shi, Lei and Feng, Yun-Long and Zhou, Ding-Xuan. Concentration estimates for learning with ℓ1-regularizer and data dependent hypothesis spaces. Applied and computational harmonic analysis

    work page

  46. [46]

    Learning with sample dependent hypothesis spaces

    Wu, Qiang and Zhou, Ding-Xuan. Learning with sample dependent hypothesis spaces. Computers & mathematics with applications (Oxford, England: 1987)

    work page 1987

  47. [47]

    Machine learning with data dependent hypothesis classes

    Cannon, A and Ettinger, J M and Hush, D and Scovel, C. Machine learning with data dependent hypothesis classes. The Journal of Machine Learning Research

    work page

  48. [48]

    Regularized modal regression with data-dependent hypothesis spaces

    Wang, Yingjie and Chen, Hong and Song, Biqin and Li, Han. Regularized modal regression with data-dependent hypothesis spaces. International Journal of Wavelets, Multiresolution and Information Processing

    work page

  49. [49]

    The eigenlearning framework: A conservation law perspective on kernel regression and wide neural networks

    Simon, James B and Dickens, Madeline and Karkada, Dhruva and DeWeese, Michael R. The eigenlearning framework: A conservation law perspective on kernel regression and wide neural networks. arXiv [cs.LG]

    work page

  50. [50]

    Frame Theory in Data Science

    Zhang, Zhihua and Jorgensen, Palle E T. Frame Theory in Data Science

    work page

  51. [51]

    Kernel Alignment Risk Estimator: Risk prediction from training data

    Jacot, Arthur and cSimcsek, Berfin and Spadaro, Francesco and Hongler, Clément and Gabriel, Franck. Kernel Alignment Risk Estimator: Risk prediction from training data. Neural Information Processing Systems

    work page

  52. [52]

    Group invariant scattering

    Mallat, Stéphane. Group invariant scattering. Communications on pure and applied mathematics

    work page

  53. [53]

    Towards understanding the spectral bias of deep learning

    Cao, Yuan and Fang, Zhiying and Wu, Yue and Zhou, Ding-Xuan and Gu, Quanquan. Towards understanding the spectral bias of deep learning. arXiv [cs.LG]

    work page

  54. [54]

    SimpleMKKM: Simple Multiple Kernel K-means

    Liu, Xinwang and Zhu, En and Liu, Jiyuan. SimpleMKKM: Simple Multiple Kernel K-means. arXiv [cs.LG]

    work page

  55. [55]

    Deep unsupervised domain adaptation: A review of recent advances and perspectives

    Liu, Xiaofeng and Yoo, Chaehwa and Xing, Fangxu and Oh, Hyejin and El Fakhri, Georges and Kang, Je-Won and Woo, Jonghye. Deep unsupervised domain adaptation: A review of recent advances and perspectives. APSIPA transactions on signal and information processing

    work page

  56. [56]

    Optimal Transport for Domain Adaptation

    Courty, Nicolas and Flamary, Rémi and Tuia, Devis and Rakotomamonjy, Alain. Optimal Transport for Domain Adaptation. arXiv [cs.LG]

    work page

  57. [57]

    Ahlberg, E N Nilson and (Eds.), J L Walsh

    J.H. Ahlberg, E N Nilson and (Eds.), J L Walsh. The Theory of Splines and Their Applications

    work page

  58. [58]

    Introduction to Fourier Analysis and Wavelets

    Pinsky, Mark A. Introduction to Fourier Analysis and Wavelets

    work page

  59. [59]

    Data Assimilation in Reduced Modeling

    Binev, Peter and Cohen, Albert and Dahmen, Wolfgang and DeVore, Ronald and Petrova, Guergana and Wojtaszczyk, Przemyslaw. Data Assimilation in Reduced Modeling. arXiv [math.NA]

    work page

  60. [60]

    Instances of computational optimal recovery: Refined approximability models

    Foucart, Simon. Instances of computational optimal recovery: Refined approximability models. Journal of complexity

    work page

  61. [61]

    Overparametrized regression: Ridgeless interpolation

    Tibshirani, Ryan. Overparametrized regression: Ridgeless interpolation. Lecture Notes from Course of Advanced Topics in Statistical Learning, Spring

    work page

  62. [62]

    Introduction to Shannon sampling and interpolation theory

    Marks, II, Robert J. Introduction to Shannon sampling and interpolation theory

    work page

  63. [63]

    The Joys of Haar Measure

    Diestel, Joe and Spalsbury, Angela. The Joys of Haar Measure

    work page

  64. [64]

    Identifiability Conditions for Domain Adaptation

    Gulrajani, Ishaan and Hashimoto, Tatsunori. Identifiability Conditions for Domain Adaptation. Proceedings of the 39th International Conference on Machine Learning

    work page

  65. [65]

    A Generalized Representer Theorem

    Schölkopf, Bernhard and Herbrich, Ralf and Smola, Alex J. A Generalized Representer Theorem. Lecture Notes in Computer Science

    work page

  66. [66]

    Modern umbral calculus: An elementary introduction with applications to linear interpolation and operator approximation theory

    Costabile, Francesco Aldo. Modern umbral calculus: An elementary introduction with applications to linear interpolation and operator approximation theory

    work page

  67. [67]

    Kernel properties and representations of boundary integral operators

    Schwab, C and Wendland, W L. Kernel properties and representations of boundary integral operators. Mathematische Nachrichten

    work page

  68. [68]

    Some properties of regularized kernel methods

    Vito, E and Rosasco, L and Caponnetto, A and Piana, M and Verri, A. Some properties of regularized kernel methods. Journal of machine learning research: JMLR

    work page

  69. [69]

    Kwok, Ivor W Tsang

    James T. Kwok, Ivor W Tsang. Learning with Idealized Kernels

    work page

  70. [70]

    Improving support vector machine classifiers by modifying kernel functions

    Amari, S and Wu, S. Improving support vector machine classifiers by modifying kernel functions. Neural networks: the official journal of the International Neural Network Society

    work page

  71. [71]

    Learning low-rank kernel matrices

    Kulis, Brian and Sustik, Mátyás and Dhillon, Inderjit. Learning low-rank kernel matrices. Proceedings of the 23rd international conference on Machine learning - ICML '06

    work page

  72. [72]

    Efficient SVM training using low-rank kernel representations

    Fine, Shai and Scheinberg, K. Efficient SVM training using low-rank kernel representations. Journal of machine learning research: JMLR

    work page

  73. [73]

    Approximating the Covariance Matrix with Low-rank Perturbations

    Magdon-Ismail, Malik and Purnell, Jonathan T. Approximating the Covariance Matrix with Low-rank Perturbations

    work page

  74. [74]

    Kristiadi-Being Bayesian, Even Just a Bit, Fixe\_supplement\_1

    work page

  75. [75]

    Analysis of Representations for Domain Adaptation

    Ben-David, Shai and Blitzer, John and Crammer, Koby and Pereira, Fernando. Analysis of Representations for Domain Adaptation. Advances in Neural Information Processing Systems

    work page

  76. [76]

    On the difficulty of approximately maximizing agreements

    Ben-David, Shai and Eiron, Nadav and Long, Philip M. On the difficulty of approximately maximizing agreements. Journal of computer and system sciences

    work page

  77. [77]

    Machine learning and the physical sciences

    Carleo, Giuseppe and Cirac, Ignacio and Cranmer, Kyle and Daudet, Laurent and Schuld, Maria and Tishby, Naftali and Vogt-Maranto, Leslie and Zdeborová, Lenka. Machine learning and the physical sciences. Reviews of modern physics

    work page

  78. [78]

    Mathematics Stack Exchange

    Eigenvalues for a product of matrices. Mathematics Stack Exchange

    work page

  79. [79]

    Eigenvalue inequalities for matrix product

    Zhang, F and Zhang, Q. Eigenvalue inequalities for matrix product. IEEE transactions on automatic control

    work page

  80. [80]

    Extreme eigenvalue distributions of some complex correlated non-central Wishart and gamma-Wishart random matrices

    Dharmawansa, Prathapasinghe and McKay, Matthew R. Extreme eigenvalue distributions of some complex correlated non-central Wishart and gamma-Wishart random matrices. Journal of multivariate analysis

    work page

Showing first 80 references.