Analyzing the structure of sampled features from an input data distribution is challenging when constrained by limited measurements in both the number of inputs and features. Traditional approaches often rely on the eigenvalue spectrum of the sample covariance matrix derived from finite measurement matrices; however, these spectra are sensitive to the size of the measurement matrix, leading to biased insights. In this paper, we introduce a novel algorithm that provides unbiased estimates of the spectral moments of the kernel integral operator in the limit of infinite inputs and features from finitely sampled measurement matrices. Our method, based on dynamic programming, is efficient and capable of estimating the moments of the operator spectrum. We demonstrate the accuracy of our estimator on radial basis function (RBF) kernels, highlighting its consistency with the theoretical spectra. Furthermore, we showcase the practical utility and robustness of our method in understanding the geometry of learned representations in neural networks.

Reproducing kernel Hilbert spaces (RKHSs) are special Hilbert spaces in one-to-one correspondence with positive definite maps called kernels. They are widely employed in machine learning to reconstruct unknown functions from sparse and noisy data. In the last two decades, a subclass known as stable RKHSs has been also introduced in the setting of linear system identification. Stable RKHSs contain only absolutely integrable impulse responses over the positive real line. Hence, they can be adopted as hypothesis spaces to estimate linear, time-invariant and BIBO stable dynamic systems from input-output data. Necessary and sufficient conditions for RKHS stability are available in the literature and it is known that kernel absolute integrability implies stability. Working in discrete-time, in a recent work we have proved that this latter condition is only sufficient. Working in continuous-time, it is the purpose of this note to prove that the same result holds also for Mercer kernels.

Reproducing kernel Hilbert spaces (RKHSs) are special Hilbert spaces where all the evaluation functionals are linear and bounded. They are in one-to-one correspondence with positive definite maps called kernels. Stable RKHSs enjoy the additional property of containing only functions and absolutely integrable. Necessary and sufficient conditions for RKHS stability are known in the literature: the integral operator induced by the kernel must be bounded as map between $\mathcal{L}_{\infty}$, the space of essentially bounded (test) functions, and $\mathcal{L}_1$, the space of absolutely integrable functions. Considering Mercer (continuous) kernels in continuous-time and the entire discrete-time class, we show that the stability test can be reduced to the study of the kernel operator over test functions which assume (almost everywhere) only the values $\pm 1$. They represent the same functions needed to investigate stability of any single element in the RKHS. In this way, the RKHS stability test becomes an elegant generalization of a straightforward result concerning Bounded-Input Bounded-Output (BIBO) stability of a single linear time-invariant system.

The last decade brought voluminous amount of work on the design, analysis and experi- mentation of kernel machines. Algorithm based on kernels can be used for various ma- chine learning tasks such as classification, regression, ranking, and principle component analysis. The most prominent learning algorithm that employs kernels is the Support Vec- tor Machines (SVM) [1, 2] designed for classification and regression. A key component in a kernel machine is a kernel operator which computes for any pair of instances their inner-product in some abstract vector space. Intuitively and informally, a kernel operator is a means for measuring similarity between instances.

The random feature model exhibits a kind of resonance behavior when the number of parameters is close to the training sample size. This behavior is characterized by the appearance of large generalization gap, and is due to the occurrence of very small eigenvalues for the associated Gram matrix. In this paper, we examine the dynamic behavior of the gradient descent algorithm in this regime. We show, both theoretically and experimentally, that there is a dynamic self-correction mechanism at work: The larger the eventual generalization gap, the slower it develops, both because of the small eigenvalues. This gives us ample time to stop the training process and obtain solutions with good generalization property.

Positive definite kernels and their associated Reproducing Kernel Hilbert Spaces provide a mathematically compelling and practically competitive framework for learning from data. In this paper we take the approximation theory point of view to explore various aspects of smooth kernels related to their inferential properties. We analyze eigenvalue decay of kernels operators and matrices, properties of eigenfunctions/eigenvectors and "Fourier" coefficients of functions in the kernel space restricted to a discrete set of data points. We also investigate the fitting capacity of kernels, giving explicit bounds on the fat shattering dimension of the balls in Reproducing Kernel Hilbert spaces. Interestingly, the same properties that make kernels very effective approximators for functions in their "native" kernel space, also limit their capacity to represent arbitrary functions. We discuss various implications, including those for gradient descent type methods. It is important to note that most of our bounds are measure independent. Moreover, at least in moderate dimension, the bounds for eigenvalues are much tighter than the bounds which can be obtained from the usual matrix concentration results. For example, we see that the eigenvalues of kernel matrices show nearly exponential decay with constants depending only on the kernel and the domain. We call this "approximation beats concentration" phenomenon as even when the data are sampled from a probability distribution, some of their aspects are better understood in terms of approximation theory.

Inner-product operators, often referred to as kernels in statistical learning, define a mapping from some input space into a feature space. The focus of this paper is the construction of biologically-motivated kernels for cortical activities. The kernels we derive, termed Spikernels, map spike count sequences into an abstract vector space in which we can perform various prediction tasks. We discuss in detail the derivation of Spikernels and describe an efficient algorithm for computing their value on any two sequences of neural population spike counts. We demonstrate the merits of our modeling approach using the Spikernel and various standard kernels for the task of predicting hand movement velocities from cortical recordings. In all of our experiments all the kernels we tested outperform the standard scalar product used in regression with the Spikernel consistently achieving the best performance.

The focus of the paper is the problem of learning kernel operators from empirical data. We cast the kernel design problem as the construction of an accurate kernel from simple (and less accurate) base kernels. We use the boosting paradigm to perform the kernel construction process. To do so, we modify the booster so as to accommodate kernel operators. We also devise an efficient weak-learner for simple kernels that is based on generalized eigen vector decomposition. We demonstrate the effectiveness of our approach on synthetic data and on the USPS dataset. On the USPS dataset, the performance of the Perceptron algorithm with learned kernels is systematically better than a fixed RBF kernel.

Inner-product operators, often referred to as kernels in statistical learning, define a mapping from some input space into a feature space. The focus of this paper is the construction of biologically-motivated kernels for cortical activities. The kernels we derive, termed Spikernels, map spike count sequences into an abstract vector space in which we can perform various prediction tasks. We discuss in detail the derivation of Spikernels and describe an efficient algorithm for computing their value on any two sequences of neural population spike counts. We demonstrate the merits of our modeling approach using the Spikernel and various standard kernels for the task of predicting hand movement velocities from cortical recordings. In all of our experiments all the kernels we tested outperform the standard scalar product used in regression with the Spikernel consistently achieving the best performance.

The focus of the paper is the problem of learning kernel operators from empirical data. We cast the kernel design problem as the construction of an accurate kernel from simple (and less accurate) base kernels. We use the boosting paradigm to perform the kernel construction process. To do so, we modify the booster so as to accommodate kernel operators. We also devise an efficient weak-learner for simple kernels that is based on generalized eigen vector decomposition. We demonstrate the effectiveness of our approach on synthetic data and on the USPS dataset. On the USPS dataset, the performance of the Perceptron algorithm with learned kernels is systematically better than a fixed RBF kernel.