Relating Leverage Scores and Density using Regularized Christoffel Functions

Relating Leverage Scores and Density
using Regularized Christoffel Functions

Edouard Pauwels, Francis Bach, Jean-Philippe Vert IRIT, Université Toulouse 3 Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France.INRIA - Département d’informatique de l’ENS Ecole normale supérieure, CNRS, INRIA PSL Research University, 75005 Paris, France.MINES ParisTech, PSL Research University, CBIO - Centre for Computational Biology, F-75006 Paris, France. Institut Curie, PSL Research University, INSERM, U900, F-75005 Paris, France. Ecole Normale Supérieure, Department of Mathematics and Applications, CNRS, PSL Research University, F-75005 Paris, France.
July 13, 2019
Abstract

Statistical leverage scores emerged as a fundamental tool for matrix sketching and column sampling with applications to low rank approximation, regression, random feature learning and quadrature. Yet, the very nature of this quantity is barely understood. Borrowing ideas from the orthogonal polynomial literature, we introduce the regularized Christoffel function associated to a positive definite kernel. This uncovers a variational formulation for leverage scores for kernel methods and allows to elucidate their relationships with the chosen kernel as well as population density. Our main result quantitatively describes a decreasing relation between leverage score and population density for a broad class of kernels on Euclidean spaces. Numerical simulations support our findings.

1 Introduction

Statistical leverage scores have been historically used as a diagnosis tool for linear regression [14, 32, 9]. To be concrete, for a ridge regression problem with design matrix and regularization parameter , the leverage score of each data point is given by the diagonal elements of . These leverage scores characterize the importance of the corresponding observations and are key to efficient subsampling with optimal approximation guarantees.

Therefore, leverage scores emerged as a fundamental tool for matrix sketching and column sampling [20, 19, 11, 34], and play an important role in low rank matrix approximation [10, 5], regression [2, 26, 18], random feature learning [27] and quadrature [6]. The notion of leverage score is seen as an intrinsic, setting-dependent quantity, and should eventually be estimated. In this work we elucidate a relation between leverage score and the learning setting (population measure and statistical model) when used with kernel methods.

For that purpose, we introduce a variant of the Christoffel function, a classical tool in polynomial algebra which provides a bound for the evaluation at a given point of a given degree polynomial in terms of an average value of . The Christoffel function is an important object in the theory of orthogonal polynomials [30, 12] and found applications in approximation theory [24] and spectral analysis of random matrices [4]. It is parametrized by the degree of polynomials considered and an associated measure, and we know that, as the polynomial degree increases, it encodes information about the support and the density of its associated measures, see [21, 22, 31] for the univariate case and [7, 8, 35, 36, 16, 17] in the multivariate case.

The variant we propose amounts to replacing the set of polynomials with fixed degree, used in the definition of the Christoffel function, by a set of function with bounded norm in a reproducing kernel Hilbert space (RKHS). More precisely, given a density on and a regularization parameter , we introduce , the regularized Christoffel function where plays a similar role as the degree for polynomials. The function turns out to have intrinsic connections with statistical leverage scores, as the quantity corresponds precisely to a notion of leverage used in [5, 2, 26, 6]. As a consequence, we uncover a variational formulation for leverage scores which helps elucidate connections with the RKHS and the density on .

Our main contribution is a precise asymptotic expansion of as under restrictions on the RKHS. To give a concrete example, if we consider the Sobolev space of functions on with squared integrable derivatives of order up to , we obtain, the asymptotic equivalent

for a continuity point of with . Here is an explicit constant which only depends on the RKHS. We recover scalings with respect to which matches known estimates for the usual degrees of freedom [26, 6]. More importantly, we also obtain a precise spatial description of (i.e., dependency on ), and deduce that the leverage score is itself proportional to in the limit. Roughly speaking, large scores are given to low density regions (note that ). This result has several potential consequences for machine learning:

(i) The Christoffel function could be used for density or support estimation. This has connections with the spectral approach proposed in [33] for support learning. (ii) This could provide a more efficient way to estimate leverage scores through density estimation. (iii) When leverage scores are used for sampling, the required sample size depends on the ratio between the maximum and the average leverage scores [26, 6]. Our results imply that this ratio can be large if there exists low-density regions, while it remains bounded otherwise.

Organization of the paper.

We introduce the regularized Christoffel function in Section 2 and explicit connections with leverage scores and orthogonal polynomials. Our main result and assumptions are described in abstract form in Section 3, they are presented as a general recipe to compute asymptotic expansions for the regularized Christoffel function. Section 3.3 describes an explicit example and a precise asymptotic for an important class of RKHS related to Sobolev spaces. We illustrate our results numerically in Section 4. The proofs are postponed to Appendix B while Appendix A contains additional properties and simulations and Appendix C contains further lemmas.

Notations.

Let denotes the ambient dimension and denote respectively the complex-valued continuous, absolutely integrable and square integrable functions over . For any , let be its Fourier transform, . For , its inverse Fourier transform is . If and , then inverse transform composed with direct transform leaves unchanged. The Fourier transform is extended to by a density argument. It defines an isometry: if , Parseval formula writes . See, e.g., [15, Chapter 11].

We identify with a set of real variables . We associate to a multi-index the monomial which degree is . The linear span of monomials form the set of -variate polynomials. The degree of a polynomial is the highest of the degrees of its monomials with nonzero coefficients (null for the null polynomial). A polynomial is said to be homogeneous of degree if for all , , , it is then composed only of monomials of degree . See [12] for further details.

2 Regularized Christoffel function

2.1 Definition

In what follows, is a positive definite, continuous, bounded, integrable, real-valued kernel on and is an integrable real function over . We denote by the RKHS associated to which is assumed to be dense in , the normed space of functions, , such that . This will be made more precise in Section 3.1.

Definition 1

The regularized Christoffel function, is given for any , by

(1)

If there is no confusion about the kernel and the density , we will use the notation . More compactly, setting for , , we have . The value of (1) is intuitively connected to the density . Indeed, the constraint forces to remains far from zero in a neighborhood of . Increasing the measure of this neighborhood increases the value of the Christoffel function. In low density regions, the constraint has little effect which allows to consider smoother functions with little overlap with higher density regions and decreases the overall cost. An illustration is given in Figure 1.

Figure 1: The black lines represent a density and the corresponding Christoffel function. The colored lines are solutions of problem in (1), the corresponding being represented by the dots. Outside the support, the optimum is smooth and high values have small overlap with the support. Inside the support, the optimum is less smooth, it forms a peak, sharper in higher density regions.

2.2 Relation with orthogonal polynomials.

The name Christoffel is borrowed from the orthogonal polynomial literature [30, 12, 24]. In this context, the Christoffel function is defined as follows for any degree :

where denotes the set of variate polynomials of degree at most . The regularized Christoffel function in (1) is a direct extension, replacing the polynomials of increasing degree by functions in a RKHS with increasing norm. has connections with quadrature and interpolation [24], potential theory and random matrices [4], orthogonal polynomials [24, 12]. Relating the asymptotic for large and properties of has also been a long lasting subject of research [21, 22, 31, 7, 8, 35, 36, 16, 17]. The idea of studying the relation between and the was directly inspired by these works.

2.3 Relation with leverage scores for kernel methods

The (non-centered) covariance of on is the bilinear form given by:

The covariance operator is then defined such that for all , If is bounded with respect to , then Lemma 5 in Appendix C shows that:

which provides a direct link with leverage scores [6], as is exactly the inverse of the population leverage score at .

As , we typically have It is worth emphasizing that spectral estimators (with other functions of the covariance operator than ) have been proposed for support inference in [33]. An example of such estimator has the form , for which finite level sets encode information about the support of  as [33]. Our main result should extend to broader classes of spectral functions.

2.4 Estimation from a discrete measure

Practical computation of the regularized Christoffel function requires to have access to the covariance operator , which is not available in closed form in general. A plugin solution consists in replacing integration with weight by a discrete approximation of the form , where for each , is a weight, and denotes the Dirac measure at . We may assume without loss of generality that the points are distinct. Given a kernel function on , let be the Gram matrix and the -th column of for . We have a closed form expression for the Christoffel function with plug-in measure , for any , :

(2)

This is a consequence of the representer theorem [28]. Note that if for all , then the Christoffel function may be obtained as a weighted diagonal element of a smoothing matrix, as for all , thanks to the matrix inversion lemma, . This draws an important connection with statistical leverage score [19, 11] as it corresponds to the notion introduced for kernel ridge regression [5, 2, 26]. It remains to choose so that approximate integration with weight .

Monte Carlo approximation:

Assuming that , if one has the possibility to draw an i.i.d. sample , with density , then one can use for . The quality of this approximation is of order (see Appendix A). If is large enough, then we obtain a good estimation of the Christoffel function (note that better bounds could be obtained with respect to using tools from [5, 2, 26]).

Riemann sums:

If the density is piecewise smooth, one can approximate integrals with weight by using a uniform grid and a Riemann sum with weights proportional to . The bound in Eq. (8) also holds, the quality of this approximation is typically of the order of which is attractive in dimension 1 but quickly degrades in larger dimensions.

3 Relating regularized Christoffel functions to density

We first precise our notations and assumptions in Section 3.1 and describe our main result in Section 3.2 using Assumption 2 which is given in abstract form. We then describe how this assumption is satisfied by a broad class of kernels in Section 3.3.

3.1 Assumptions

Assumption 1
  1. The kernel is translation invariant: for any , where is the inverse Fourier transform of which is real valued and strictly positive.

  2. The density is finite and nonnegative everywhere.

Under Assumption 1, is a positive definite kernel by Bochner’s theorem and we have an explicit characterization of the associated RKHS (see e.g. [33, Proposition 4]),

(3)

with inner product

(4)
Remark 1

The assumption that implies by the Riemann-Lebesgue theorem that is in , the set of continuous functions vanishing at infinity. Since is strictly positive, its support is  and [29, Proposition 8] implies that is -universal, i.e., that is dense in w.r.t. the uniform norm. As a result, is also dense in for any probability measure .

Remark 2

For any , we have by Cauchy-Schwartz inequality

and the last term is finite by Assumption 1. Hence and we have where the integral is understood in the usual sense. In this setting any is uniquely determined everywhere on by its Fourier transform and we have for any , .

3.2 Main result

Problem (1) is related to a simpler variational problem with explicit solution. For any , let

(5)

Note that does not depend on and corresponds to the Christoffel function at (or any other points by translation invariance) for the Lebesgue measure on . The solutions of (5) have an explicit description which proof is presented in Appendix B.2.

Lemma 1

For any , , and this value is attained by the function

Remark 3

We directly obtain , for any . Finally, let us mention that Assumption 1 ensures that as which diverges as .

We denote by the inverse Fourier transform of , i.e., . It satisfies . Intuitively, as tends to , , should be approaching a Dirac in the sense that tends to everywhere except at the origin where it goes to . The purpose of the next Assumption is to quantify this intuition.

Assumption 2

For the kernel given in Assumption 1 and given in Lemma 1, there exists such that, as , , and

See Section 3.3 for specific examples. We are now ready to describe the asymptotic inside the support of , the proof is given in Appendix B.1.

Theorem 1

Let , and be given as in Assumption 1 and let be defined as in (1). If Assumption 2 holds, then, for any such that and is continuous at , we have

Proof sketch.

The equivalent is shown by using the variational formulation in Eq. (1). A natural candidate for the optimal function is the optimizer obtained from Lebesgue measure in Eq. (5), scaled by . Together with Assumption 2, this leads to the desired upper bound. In order to obtain the corresponding lower bound, we consider Lebesgue measure restricted to a small ball around . Using linear algebra and expansions of operator inverses, we relate the optimal value directly to the optimal value of Eq. (5).

This result is complemented by the following which describes the asymptotic behavior outside the support of , the proof is given in Appendix B.3.

Theorem 2

Let , and be given as in Theorem 1. Then, for any , such that there exists with , we have

If furthermore there exists and such that, for any , , then, for any such , we have

Proof sketch.

Since only an upper-bound is needed, we simply have to propose a candidate function for , and we build one from the solution of Eq. (5) for (i) and directly from properties of kernels for (ii).

Remark 4

Theorem 1 and 2 underline separation between the “inside” and the “outside” of the support of and describes the fact that the convergence to as decreases is faster outside: (i), if with (which is the case in most interesting situations), then . (ii), it holds that . Hence in most cases, the values of the Christoffel function outside of the support of are negligible compared to the ones inside the support of .

Combining Theorem 1 and 2 does not describe what happens in the limit case where neither of the conditions on hold, for example on the boundary of the support or at discontinuity points of the density. We expect that this highly depends on the geometry of and its support. In the polynomial case on the simplex, the rate depends on the dimension of the largest face containing the point of interest [36]. Settling down this question in the RKHS setting is left for future research.

3.3 A general construction

We describe a class of kernels for which Assumptions 1 and 2 hold, and Theorem 1 can be applied, which includes Sobolev spaces. We also compute explicit equivalents for in (5). We first introduce a definition and an assumption.

Definition 2

For any , a -variate polynomial of degree is called -positive if it satisfies the following.

  • Let denote the -homogeneous part of (the sum of its monomial of degree ). is (strictly) positive on the unit sphere in .

  • The polynomial satisfies for all .

Remark 5

If is -positive, then it is always greater than and its -homogeneous part is strictly positive except at the origin. The positivity of forbids the use of polynomial of the form which would allow to treat product kernels. Indeed, this would lead to which is not positive on the unit sphere. The last condition on is not very restrictive as it can be ensured by a proper rescaling of if we have only.

Assumption 3

Let be a -positive, -variate polynomial and let be such that . The kernel is given as in Assumption 1 with .

One can check that in Assumption 3 is well defined and satisfies Assumption 1. A famous example of such a kernel is the Laplace kernel which amounts, up to a rescaling, to choose of the form for and . In addition, Assumption 3 allows to capture the usual multi-dimensional Sobolev space of functions with square integrable partial derivatives up to order , with , and the corresponding norm. We now provide the main result of this section.

Lemma 2

Assume that and are given as in Assumption 1 and 3. Then Assumption 2 is satisfied. More precisely, set and , then for any the following holds true as , :

(i)
Remark 6

If , using spherical coordinate integration, we obtain

The proof is presented in Appendix B.4. We have the following corollary which is a direct application of Theorem 1. It explicits the asymptotic for the Christoffel function, in terms of the density .

Corollary 1

Assume that and are given as in Assumption 1 and 3 and that is such that and is continuous at . Then as , ,

4 Numerical illustration

In this section we provide numerical evidences confirming the rate described in Corollary 1. We use the Matérn kernel, a parametric radial kernel allowing different values of in Assumption 3.

4.1 Matérn kernel

We follow the description of [25, Section 4.2.1], note that the Fourier transform is normalized differently in our paper. For any and , we let for any ,

(6)

where is the modified Bessel function of the second kind [1, Section 9.6]. This choice of satisfies Assumption 3, with and . Indeed, for any , its Fourier transform is given for any

(7)

4.2 Empirical validation of the convergence rate estimate

Corollary 1 ensures that, given and in (6), as , we have for appropriate , . We use the Riemann sum plug-in approximation described in Section 2.4 to illustrate this result numerically. We perform extensive investigations with compactly supported sinusoidal density in dimension 1. Note that from Remark 6 we have the closed form expression .

Figure 2: The target density represented in red. We consider different choices of and for as in (6). We use the Riemann sum plug-in approximation described in (2) with . Left: the fact that the estimate is close to the density is clear for small values of . Right: the dotted line represents the identity. This suggests that the rate estimate is of the correct order in .

Relation with the density:

For a given choice of , as , we should obtain for appropriate that the quantity, is roughly equal to . This is confirmed numerically as presented in Figure 2 (left), for different choices of the parameters .

Convergence rate:

For a given choice of , as , we should obtain for appropriate that the quantity is roughly equal to . Considering the same experiment confirms this finding as presented in Figure 2 right, which suggests that the exponent in is of the correct order.

Additional experiments:

A piecewise constant density is considered in Appendix A which also contains simulations suggesting that the asymptotic has a different nature for the Gaussian kernel for which we conjecture that our result does not hold.

5 Conclusion and future work

We have introduced a notion of Christoffel function in RKHS settings. This allowed to derive precise asymptotic expansion for a quantity known as statistical leverage score which has a wide variety of applications in machine learning with kernel methods. Our main result states that the leverage score is inversely proportional to a power of the population density at the considered point. This has intuitive meaning as leverage score is a measure of the contribution of a given observation to a statistical estimate. For densely populated region, a specific observation, which should have many close neighbors, has less effect on a statistical estimate than observations in less populated areas of space. Our observation gives a precise meaning to this statement and sheds new light on the relevance of the notion of leverage score. Furthermore, it is coherent with known results in the orthogonal polynomial literature from which the notion of Christoffel function was inspired.

Direct extensions of this work include approximation bounds for our proposed plug-in estimate and tuning of the regularization parameter . A related question is about the relevance of the proposed variational formulation for the statistical estimation of leverage scores, learning from random features, in particular random Fourier features and density/support estimation. Another line of future research would be the extension of our estimates to broader classes of RKHS, for example, kernels with product structure, such as the counterpart of the Laplace kernel. Finally, it would be interesting to extend the concepts to abstract topological spaces beyond .

Acknowledgement

We acknowledge support from the European Research Council (grant SEQUOIA 724063).

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Appendix A Additional properties and numerical simulations

Monotonicity properties.

It is obvious from the definition in (1) that the regularized Christoffel function is an increasing function of , it is also concave. If and are as in Assumption 1.2, for any ,

that is, the regularized Christoffel function is an increasing function of the underlying density.

The Christoffel function is also monotonic with respect to kernel choice. For any two positive definite kernels and , we have for any ,

that is, the regularized Christoffel function is a decreasing function of the underlying kernel. Indeed, for any positive definite kernels and , denote by the RKHS associated to and the RKHS associated to . We have and [3, Section 7, Theorem I].

Overfitting.

We are interested in the asymptotic behavior of the Christoffel function as the regularization parameter tends to . This is approximated based on points using the plug-in approach in Section 2.4. For a fixed value of , the empirical measure is supported on only points and the asymptotic as is straightforward. For example if Theorem 2 (ii) holds, then we obtain outside of the support and at each support point , . This is because the quality of approximation of by depends on the regularity of the corresponding test functions. Small values of the regularization parameter allow to consider functions with very low regularity so that the approximation become vacuous and the obtained estimate only reflects the finiteness of the support of . This phenomenon is illustrated in Figure 3. Hence, when using the proposed plug-in approach, it is fundamental to carefully tune the considered value of as a function of . Theoretical guidelines for measuring this trade-off are left for future research, in our experiments, this is done on an empirical basis (we prove below a loose sufficient condition, where has to be large).

Figure 3: Illustration of the overfitting phenomenon. The target density is represented in red. We approximate it by supported on the black dots with corresponding weights proportional to , . For , we use Eq. (2) to compute the corresponding empirical Christoffel function represented in dark blue.

Monte Carlo approximation:

Assuming that , if one has the possibility to draw an i.i.d. sample , with density , then one can use for . Our estimators take the form , where is the empirical covariance operator. Thus, we have:

(8)

Since, is of order (see, e.g., [23]), if is large enough, then we obtain a good estimation of the Christoffel function (note that better bounds could be obtained with respect to using tools from [5, 2, 26]).

Gaussian kernel:

A natural question is whether Theorem 1 holds for the Gaussian kernel where is a bandwidth parameter. For this choice of kernel, is of the order of , which decreases very slowly. We conjecture that Assumption 2 fails in this setting and that Theorem 1 does not hold. Indeed, performing the same simulation as in Section 4 with a piecewise constant density, we observe that the localization phenomenon no longer holds. This is presented in Figure 4 which displays important boundary effects around discontinuities. For comparison purpose, Figure 5 gives the same result for Mattérn kernels.

Figure 4: Estimate of the Christoffel function for the Gaussian kernel, different values of the bandwidth and a piecewise constant density. The setting is the same as in the experiment presented in Section 4. The behavior at the discontinuities suggest that variations of the density affect the value of the Christoffel function beyond the local scale described in Theorem 1.
Figure 5: Estimate of the Christoffel function for the Mattèrn kernel, different values of the parameters and a piecewise constant density. The setting is the same as in the experiment presented in Section 4. The behavior at the discontinuities suggest that variations of the density affect the value of the Christoffel only at the local scale described in Theorem 1.

Illustration in dimension 2:

For illustration purpose, we consider a density on the unit square in dimension 2 and compare it with the estimate obtained from the regularized Christoffel function using the Riemann plug-in approximation procedure. We choose the Matérn kernel with and and . Figure 6 illustrates the correspondence between the true density and the obtained estimate.

Figure 6: Comparison between the level sets of a given density (right) and the estimate given by the empirical Christoffel function with , and (left). We use the Riemann plug-in approximation procedure with a grid of 2025 points on . The estimate captures both the round shape of the level sets in the middle and the squared shape of the support ().

Appendix B Proofs

b.1 Proof of Theorem 1

The proof is organized as follows, first we will prove an upper bound on which is of the order of the claimed equivalent plus negligible terms. In a second step we produce a lower bound on which is of the same nature. Assumptions 1 and 2 are assumed to hold true throughout this section.

Recall that we have with the notations given in Eq. (3) of the main text. We will work with since the general result can be obtained by a simple translation. We consider as in Assumption 1 and assume throughout the section that . This is without loss of generality since , one can substitute by and by and use

(9)

Combining translations and scaling in (9), we only need to show that when and is continuous at .

Upper bound:

For any , is feasible for in problem (1) and therefore, using ,

(10)

We only need to control the last term. The result then follows from the next Lemma which proof is postponed to Section B.2.

Lemma 3

As with ,

Combining (10) and Lemma 3, we obtain, as

(11)

Lower bound:

To prove the lower bound, let . The quantity is non negative and we have by continuity of . Choosing as given by Assumption 2, we obtain for any sufficiently small, using ,

(12)

We need to control the last term. This is the purpose of the following Lemma which proof is postponed to Section B.2.

Lemma 4

Let be given as in Assumption 2, then, as with , we have

Combining (12) and Lemma 4, we obtain, as

(13)

To conclude, combining (11) and (13), we obtain as claimed.

b.2 Lemmas for Section b.1 and proof of Lemma 1 of the main text

Proof of Lemma 1: Eq. (3) characterizes and in particular, any function in is in so that Parseval theorem holds. Furthermore for any , is in (see Remark 2). Rewriting (5) in the Fourier domain, we have

s.t.
(14)

The space endowed with the inner product is a Hilbert space which is simply the image of by the Fourier transform. Problem (14) can be rewritten in a form that fits Lemma 5 below as follows

s.t.
(15)

where is the operator which consists in multiplication by . For any , we have and is bounded on . Using Lemma 5, we get an expression for the solution of the minimization problem in (14) of the form

for all , where the optimal value , ensures that . We deduce the value of and get back to by combining Eq. (3) with the inverse Fourier transform of which leads to the claimed expression for .  

Proof of Lemma 3: Let . We have by continuity of . Let be given as in Assumption 2.

Using Assumption 2, as , the first term is and the sum is also . This proves the desired result.  

Proof of Lemma 4: Consider the surroga