Optimal Recovery of Local Truth

C. C. RODRIGUEZ

Abstract

1  Introduction

During the past thirty years the theory of Nonparametrics has been dominating the scene in mathematical statistics. Parallel to the accelerating discovery of new technical results, a consensus has been growing among some researchers in the area, that we may be witnessing a promising solid road towards the elusive Universal Learning Machine (see e.g. [,]).

The queen of nonparametrics is density estimation. All the fundamental ideas for solving the new problems of statistical estimation in functional spaces (smoothing, generalization, optimal minimax rates, etc.) already appear in the problem of estimating the probability density (i.e. the model) from the observed data. More over, it is now well known that a solution for the density estimation problem automatically implies solutions for the problems of pattern recognition and nonparametric regression as well as for most problems that can be expressed as a functional of the density.

In this paper I present a technical result, about optimal nonparametric density estimation, that shows at least at a formal level, a surprising similarity between nonparametrics and General Relativity. Simply put,

What exactly it is meant by this is the subject of this paper but before proceeding further a few comments are in order. First of all, let us assume that we have a set {x₁,¼,x_n} of data. Each observation x_j consisting of p measurements that are thought as the p coordinates of a vector in Â^p. To make the data space into a probability space we endow Â^p with the field of Borelians but nothing beyond that. In particular no a priori metric structure on the data space is assumed. The n observations are assumed to be n independent realizations of a given probability measure P on Â^p. By the Lebesgue decomposition theorem, for every Borel set B we can write,

P(B) = ó õ B  f(x) l(dx) + n(B)

(1)

where n is the singular part of P that assigns positive probability mass to Borel sets of zero Lebesgue volume. Due to the existence of pathologies like the Cantor set in one dimension and its analogies in higher dimensions, the singular part n cannot be empirically estimated (see e.g. []). Practically all of the papers on density estimation rule out the singular part of P a priori. The elimination of singularities by fiat has permitted the construction of a rich mathematical theory for density estimation, but it has also ruled out a priori the study of models of mixed dimensionality (see []) that may be necessary for understanding point masses and spacetime singularities coexisting with absolutely continuous distributions.

We assume further that in the regions where f(x) > 0 the density f is of class C² i.e., it has continuous second order partial derivatives.

1.1  Nonparametrics with the World in Mind

The road from Classical Newtonian Physics to the physics of today can be seen as a path paved by an increasing use of fundamental concepts that are statistical in nature. This is obvious for statistical mechanics, becoming clearer for quantum theory, and appearing almost as a shock in General Relativity. Not surprisingly there have been several attempts to take this trend further (see e.g. [,,,]) in the direction of Physics as Inference.

Now suppose for a moment that in fact some kind of restatement of the foundations of physics in terms of information and statistical inference will eventually end up providing a way to advance our current understanding of nature. As of today, that is either already a solid fact or remains a wild speculation, depending on who you ask. In any case, for the trend to take over, it will have to be able to reproduce all the successes of current science and make new correct predictions. In particular it would have to reproduce General Relativity. Recall that the main lesson of General Relativity is that space and time are not just a passive stage on top of which the universe evolves. General Relativity is the theory that tells (through the field equation) how to build the stage (left hand side of the equation) from the data (right hand side of the equation). The statistical theory that tells how to build the stage of inference (the probabilistic model) from the observed data is: Nonparametric Density Estimation. It is therefore reassuring to find typical signatures of General Relativity in density estimation as this paper does. Perhaps Physics is not just a special case of statistical inference and all these are only coincidences of no more relevance than for example the fact that multiplication or the logarithmic function appear everywhere all the time. That may be so, but even in that case I believe it is worth noticing the connection for undoubtedly GR and density estimation have a common goal: The dynamic building of the stage.

More formally. Let f be a multivariate probability density function with continuous second order partial derivatives. Consider the problem of estimating the true value of f(z) > 0 at a single point z, from n independent observations. It is shown that, fastest possible estimators (including the k-nearest neighbor and kernel as well as the rich class of estimators in []) have minimum asymptotic mean square errors when the space of observations is thought as conformally curved. The optimal metric is shown to be generated by the Hessian of f in the regions where the Hessian is definite. Thus, the peaks and valleys of f are surrounded by horizons where the Hessian changes signature from Riemannian to pseudo-Riemannian.

The result for the case of generalized k-nearest neighbor estimators [] has circulated since 1988 in the form of a technical report []. Recently I found that a special case of this theorem has been known since 1972 [] and undergone continuous development in the Pattern Recognition literature, (see e.g. [,,,]).

2  Estimating Densities from Data

The canonical problem of density estimation at a point z Î Â^p can be stated as follows: Estimate f(z) > 0 from n independent observations of a random variable with density f.

The most successful estimators of f(z) attempt to approximate the density of probability at z by using the observations x₁,¼,x_n  to build both, a small volume around z and, a weight for this volume in terms of probability mass. The density is then computed as the ratio of the estimated mass over the estimated volume. The two classical examples are the k-nearest neighbor (knn) and the kernel estimators.

2.1  The knn

lk = l(S(R(k)))

(3)

where,

S(r) = { x Î Âp : ||x-z|| £ r }

(4)

The sphere S(r) and the radius R(k) are defined relative to a given norm, ||·|| in Â^p. The stochastic behavior of the knn depends on the specific value of the integer k chosen in (2). Clearly, in order to achieve consistency (e.g. stochastic convergence of h_n(z) as n®¥ towards the true value of f(z) > 0) it is necessary to choose k = k(n) as a function of n. The volumes l_k must shrink, to control the bias, and consequently we must have k/n® 0 for h_n(z) to be able to approach a strictly positive number. On the other hand, we must have k®¥ to make the estimator dependent on an increasing number k of observations and in this way to control its variance. Thus, for the knn to be consistent, we need k to increase with n but at a rate slower than n itself.

The knn estimator depends not only on k but also on a choice of norm. The main result of this paper follows from the characterization of the ||·|| that, under some regularity conditions, produces the best asymptotic (as n®¥) performance for density estimators.

2.2  The kernel

gn(z) = Mm l(S(m))

(6)

where,

Mm = 1 n æ è å xj Î S(m)  Km-1(xj-z) ö ø

(7)

The smoothing parameter m = m(n) is such that k = [nm^p] satisfies the conditions for consistency of the knn, K_m^-1(x) = K(m^-1x) where the kernel function K is a non negative bounded function with support on the unit sphere (i.e. K(x) = 0 for ||x|| > 1) and satisfying,

ó õ ||x|| £ 1  K(x) dx = b

(8)

Notice that for the constant kernel (i.e. K(x) = 1 for ||x|| £ 1) the estimator (6) approximates f(z) by the proportion of observations inside S(m) over the volume of S(m). The general kernel function K acts as a weight function allocating different weights K_m^-1(x_j-z) to the x_j's inside S(m). To control bias (see () below) the kernel K is usually taken as a decreasing radially symmetric function in the metric generated by the norm ||·||. Thus, K_m^-1(x_j-z) assigns a weight to x_j that decreases with its distance to z. This has intuitive appeal, for the observations that lie closer to z are less likely to fall off the sphere S(m), under repeated sampling, than the observations that are close to the boundary of S(m).

The performance of the kernel as an estimator for f(z) depends first and foremost on the value of the smoothness parameter m. The numerator and the denominator of g_n(z) depend not only on m but also on the norm ||·|| chosen and the form of the kernel function K. As it is shown in theorem () these three parameters are inter-related.

2.3  Double Smoothing Estimators

The knn (2) and the kernel (6) methods are two extremes of a continuum. Both, h_n(z) and g_n(z) estimate f(z) as probability-mass-per-unit-volume. The knn fixes the mass to the deterministic value k/n and lets the volume l_k to be stochastic, while the kernel method fixes the volume l(S(m)) and lets the mass M_m to be random. The continuum gap between (2) and (6) is filled up by estimators that stochastically estimate mass and volume by smoothing the contribution of each sample point with different smoothing functions for the numerator and denominator (see []).

Let b ³ 1 and assume, without loss of generality that bk is an integer. The double smoothing estimators with deterministic weights are defined as,

fn(z) = 1 n n å i = 1  K æ ç è z-xi R(k) ö ÷ ø 1 cb bk å i = 1  wi li

(9)

where,

wi = ó õ i/bk (i-1)/bk  w(u) du

(10)

and w(·) is a probability density on [0,1] with mean c.

3  The Truth as n®¥ ?

In nonparametric statistics, in order to assess the quality of an estimator f_n(z) as an estimate for f(z), it is necessary to choose a criterion for judging how far away is the estimator from what it tries to estimate. This is sometimes regarded as revolting and morally wrong by some Bayesian Fundamentalists. For once you choose a loss function and a prior, logic alone provides you with the Bayes estimator and the criterion for judging its quality. That is desirable, but there is a problem in high dimensional spaces. In infinite dimensional hypothesis spaces (i.e. in nonparametric problems) almost all priors will convince you of the wrong thing! (see e.g. [,] for a non-regular way out see []). These kind of Bayesian nonparametric results provide a mathematical proof that: almost all fundamental religions are wrong, (more data can only make the believers more sure that the wrong thing is true!). An immediate corollary is that: Subjective Bayesians can't go to Heaven. Besides, the choice of goodness of fit criterion is as ad-hoc (an equivalent) to the choice of a loss function.

3.1  The Natural Invariant Loss Function and Why the MSE is not that Bad

The most widely studied goodness of fit criterion is the Mean Square Error (MSE) defined by,

(MSE) = E|fn(z) - f(z)|2

(11)

where the expectation is over the joint distribution of the sample x₁,¼,x_n  . By adding and subtracting T = Ef_n(z) and expanding the square, we can express the MSE in the computationally convenient form,

(MSE) = E|fn(z) - T|2 + |T - f(z)|2 = (variance) + (bias)2 (12)

By integrating (11) over the z Î Â^p and interchanging E and ò (OK by Fubbini's theorem since the integrand ³ 0) we obtain,

(MISE) = E ó õ |fn(z) - f(z)|2 dz

(13)

The Mean Integrated Square Error (MISE) is just the expected L² distance of f_n from f. Goodness of fit measures based on the (MSE) have two main advantages: They are often easy to compute and they enable the rich Hilbertian geometry of L². On the other hand the (MSE) is unnatural and undesirable for two reasons: Firstly, the (MSE) is only defined for densities in L² and this rules out a priori all the densities in L¹\L² which is unacceptable. Secondly, even when the (MISE) exists, it is difficult to interpret (as a measure of distance between densities) due to its lack of invariance under relabels of the data space. Many researchers see the expected L¹ distance between densities as the natural loss function in density estimation. The L¹ distance does in fact exist for all densities and it is easy to interpret but it lacks the rich geometry generated by the availability of the inner product in L². A clean way out is to use the expected L² distance between the wave functions y_n = Ö{f_n} and y = Öf.

Theorem 1 The L² norm of wave functions is invariant under relabels of the data space, i.e.,

ó õ |yn(z) - y(z)|2 dz = ó õ | ~ y   n  (z¢) - ~ y   (z¢)|2 dz¢

(14)

where z = h(z¢) with h any one-to-one smooth function.

Proof: Just change the variables. From, the change of variables theorem the pdf of z¢ is,

~ f   (z¢) = f(h(z¢)) ê ê ê ¶(h) ¶(z¢) ê ê ê

(15)

from where the wave function of z¢ is given by,

~ y   (z¢) = y(h(z¢)) ê ê ê ¶(h) ¶(z¢) ê ê ê 1/2

(16)

Thus, making the substitution z = h(z¢) we get,

ó õ |yn - y|2 dz = ó õ |yn(h(z¢)) - y(h(z¢))|2 ê ê ê ¶(h) ¶(z¢) ê ê ê dz¢ = ó õ |yn ê ê ê ¶(h) ¶(z¢) ê ê ê 1/2   - y ê ê ê ¶(h) ¶(z¢) ê ê ê 1/2   |2 dz¢ = ó õ | ~ y   n  - ~ y   |2 dz¢ (17)

Q.E.D.

The following theorem shows that a transformation of the MSE of a consistent estimator provides an estimate for the expected L² norm between wave functions.

Theorem 2 Let f_n(z) be a consistent estimator of f(z). Then,

E ó õ |yn - y|2 dz = 1 4 ó õ E|fn(z) - f(z)|2 f(z) dz + (smaller order terms)

(18)

Proof: A first order Taylor expansion of Öx about x₀ gives,

Öx - Ö   x0   = 1 2 (x-x0) Ö x0 + o((x-x0)2)

(19)

Substituting x = f_n(z), x₀ = f(z) into (19) squaring both sides and taking expectations we obtain,

E|yn(z) - y(z)|2 = 1 4 E|fn(z)-f(z)|2 f(z) + o(E|fn(z)-f(z)|2)

(20)

integrating over z and interchanging E and ò we arrive at (18).

Q.E.D.

Proceeding as in the proof of theorem 3.1 we can show that

ó õ |fn -f|2 f dz = ó õ | ~ f   n  - ~ f   |2 ~ f dz¢

(21)

where, as before, z« z¢ is any one-to-one smooth transformation of the data space and [f\tilde] is the density of z¢. Thus, it follows from (21) that the leading term on the right hand side of (18) is also invariant under relabels of the data space. The nice thing about the L² norm of wave functions, unlike (21), is that it is defined even when f(z) = 0.

4  Some Classic Asymptotic Results

We collect here the well known Central Limit Theorems (CLT) for the knn and kernel estimators together with some remarks about nonparametric density estimation in general. The notation and the formulas introduced here will be needed for computing the main result about optimal norms in the next section.

Assumption 1 Let f be a pdf on Â^p of class C² with non singular Hessian, H(z) at z Î Â^p, and with f(z) > 0, i.e., the matrix of second order partial derivatives of f at z exists, it is non singular and its entries are continuous at z.

Assumption 2 Let K be a bounded non negative function defined on the unit sphere, S₀ = {x Î Â^p: ||x|| £ 1} and satisfying,

ó õ ||x|| £ 1  K(x) dx = l(S0) º b (22) ó õ ||x|| £ 1  x K(x) dx = 0 Î Âp (23)

Theorem 3 [CLT for knn] Under assumption 4, if k = k(n) is taken in the definition of the knn (2) in such a way that for some a > 0

lim n®¥  n-4/(p+4) k = a

(24)

then, if we let Z_n = Ök(h_n(z) - f(z)) we have,

lim n®¥  P(Zn £ t) = ó õ t -¥  1 Ö 2p exp æ ç è - (y-B(z))2 2V(z) ö ÷ ø dy

(25)

where,

B(z) = æ ç è a[(p+4)/2p] 2f2/p(z) ö ÷ ø ì í î b-1-2/p ó õ ||x|| £ 1  xTH(z)x dx ü ý þ

(26)

and,

V(z) = f2(z)

(27)

Proof: This is a special case of [].

Theorem 4 [CLT for kernel] Under assumptions 4, and 4 if m = m(n) is taken in the definition of the kernel (6) in such a way that for some a > 0, k = [nm^p] satisfies (24) then, if we let Z_n = Ök(g_n(z) - f(z)) we have (25) where now,

B(z) = æ ç è a[(p+4)/2p] 2 ö ÷ ø ì í î b-1 ó õ ||x|| £ 1  xTH(z)x K(x) dx ü ý þ

(28)

and,

V(z) = f(z) ì í î b-2 ó õ ||x|| £ 1  K2(x) dx ü ý þ

(29)

Proof: The sample x₁,¼,x_n  is assumed to be iid f and therefore the kernel estimator g_n(z) given by (6) and (7) is a sum of iid random variables. Thus, the classic CLT applies and we only need to verify the rate (24) and the asymptotic expressions for the bias (28) and variance (29). We have,

E[ gn(z) ] = 1 bmp 1 n n å j = 1  ó õ K æ ç è xj-z m ö ÷ ø f(xj) dxj (30) = 1 bmp ó õ K(y) f(z+my)mp dy (31) = ó õ K(y) b ì í î f(z) + mÑf(z)·y + m2 2 yTH(z)y + o(m2) ü ý þ dy (32) = f(z) + m2 2b ó õ yTH(z)y K(y) dy + o(m2) (33)

where we have changed the variables of integration to get (31), used assumption 4 and Taylor's theorem to get (32) and used assumption 4 to obtain (33). For the variance we have,

var(gn(z)) = 1 nb2m2p var (K((X-z)/m)) (34) = 1 nb2m2p ì í î ó õ ||y|| £ 1  f(z+my) K2(y) mpdy - æ è ó õ ||y|| £ 1  f(z+my)K(y) mpdy ö ø 2   ü ý þ (35) = f(z) nb2mp ó õ ||y|| £ 1  K2(y)dy + o æ ç è 1 nmp ö ÷ ø (36)

where we have used var(K) = EK²-(EK)² and changed the variables of integration to get (35), used assumption 4 and (0th order) Taylor's theorem to get (36). Hence, the theorem follows from (33) and (36) after noticing that (24) and k = nm^p imply,

Ökm2 = k[(4+p)/2p]n-2/p = (n-[4/(p+4)]k)[(p+4)/2p] ® a[(p+4)/2p] (37) k nmp = k k = 1 (38)

Q.E.D.

Theorem 5 [CLT for double smoothers] Consider the estimator f_n(z) defined in (9). Under assumptions 4, 4, and (24) if we let Z_n = Ök(f_n(z) - f(z)) we have (25) where now,

B(z) = æ ç è a[(p+4)/2p] 2[bf(z)]2/p ö ÷ ø b-1 ì í î ó õ ||x|| £ 1  xTH(z)x [K(x)+l0] dx ü ý þ

(39)

and,

V(z) = f2(z) ì í î b-1 ó õ ||x|| £ 1  K2(x) dx - l1 ü ý þ

(40)

with,

l0 = b2/p c ó õ 1 0  u1+[2/p] w(u) du - 1 (41) l1 = 1 - c-2b-1 ó õ 1 0  ì í î ó õ 1 y  w(x) dx ü ý þ 2   dy (42)

Proof: See []. Remember to substitute K by b^-1K since in the reference the Kernels are probability densities and in here we take them as weight functions that integrate to b.

4.1  Asymptotic Mean Square Errors

Let f_n be an arbitrary density estimator and let Z_n = Ök(f_n(z)-f(z)). Now suppose that f_n(z) is asymptotically normal, in the sense that when k = k(n) satisfies (24) for some a > 0, we have (25) true. Then, all the moments of Z_n will converge to the moments of the asymptotic Gaussian. In particular the mean and the variance of Z_n will approach B(z) and V(z) respectively. Using, (12) and (24) we can write,

lim n®¥  n4/(p+4) E|fn(z)-f(z)|2 = V(z) a + B2(z) a

(43)

We call the right hand side of (43) the asymptotic mean square error (AMSE) of the estimator f_n(z). The value of a can be optimized to obtain a global minimum for the (AMSE) but it is well known in nonparametrics that the rate n^-4/(p+4) is best possible (in a minimax sense) under the smoothness asumption 4 (see e.g. []). We can take care of the knn, the kernel, and the double smoothing estimators simultaneously by noticing that in all cases,

(AMSE) = a1a-1 + a2a4/p

(44)

has a global minimum of,

(AMSE)* = ì í î (1+4/p) æ ç è p 4 ö ÷ ø [4/(p+4)]   ü ý þ a1[4/(p+4)]a2[p/(p+4)]

(45)

achieved at,

a* = æ ç è pa1 4a2 ö ÷ ø [p/(p+4)]

(46)

Replacing the corresponding values for a₁ and a₂ for the knn, for the kernel, and for the double smoothing estimators, we obtain that in all cases,

(AMSE)* = (const. indep. of f) ì í î f(z) æ ç è D2 f(z) ö ÷ ø [p/(p+4)]   ü ý þ

(47)

where,

D = ó õ ||x|| £ 1  xTH(z)x G(x) dx (48) = p å j = 1  rj ¶2f ¶zj2 (z) (49)

with G(x) = 1 for the knn, G(x) = K(x) for the kernel, G(x) = K(x)+l₀ for the double smoothers (see (39) and (41)) and, if e_j denotes the jth canonical basis vector (all zeroes except a 1 at position j),

rj = ó õ ||x|| £ 1  (x·ej)2 G(x) dx

(50)

Notice that (49) follows from (48), (23) and the fact that H(z) is the Hessian of f at z. The generality of this result shows that (47) is typical for density estimation. Thus, when f_n is either the knn, the kernel, or one of the estimators in ( rodriguez86), we have:

lim n®¥  n4/(p+4)E|fn(z)-f(z)|2 ³ c f(z) æ ç è D2 f(z) ö ÷ ø [p/(p+4)]

(51)

The positive constant c may depend on the particular estimator but it is independent of f. Dividing both sides of (51) by f(z), integrating over z, using theorem 3.1 and interchanging E and ò we obtain,

lim n®¥  n4/(p+4)E ó õ |yn(z)-y(z)|2 dz ³ 4c ó õ ê ê ê D y(z) ê ê ê [2p/(p+4)]   dz

(52)

The worst case scenario is obtained by the model f = y² that maximizes the action given by the right hand side of (52),

L = ó õ ê ê ê 1 y(z) p å j = 1  rj ¶2y2 ¶zj2 (z) ê ê ê [2p/(p+4)]   dz

(53)

This is a hard variational problem. However, it is worth noticing that the simplest case is obtained when the exponent is 1, i.e. when the dimension of the data space is p = 4. Assuming we were able to find a solution, this solution would still depend on the p parameters r₁,¼,r_p. A choice of r_j's is equivalent to the choice of a global metric for the data space. Notice also, that the exponent becomes 2 for p = ¥ and that for p ³ 3 (but not for p = 1 or 2) there is the possibility of non trivial (i.e. different from uniform) super-efficient models for which estimation can be done at rates higher than n^-4/(p+4). These super-efficient models are characterized as the non negative solutions of the Laplace equation in the metric generated by the r_j's, i.e., non negative (f(z) ³ 0) solutions of,

p å j = 1  rj ¶2f ¶zj2 (z) = 0

(54)

Recall that there are no non trivial (different from constant) non negative super-harmonic functions in dimensions one or two but there are plenty of solutions in dimension three and higher. For example the Newtonian potentials,

f(z) = c||z||r-(p-2)

(55)

with the norm,

||z||r2 = p å j = 1  æ ç ç ç è zj Ö rj ö ÷ ÷ ÷ ø 2

(56)

will do, provided the data space is compact. The existence of (hand picked) super-efficient models is what made necessary to consider best rates only in the minimax sense. Even though we can estimate a Newtonian potential model at faster than usual nonparametric rates, in any neighborhood of the Newtonian model the worst case scenario is at best estimated at rate n^-4/(p+4) under second order smoothness conditions.

5  Choosing the Optimal Norm

All finite (p < ¥) dimensional Banach spaces are isomorphic (as Banach spaces) to Â^p with the euclidian norm. This means, among other things, that in finite dimensional vector spaces all norms generate the same topology. Hence, the spheres {x Î Â^p:||x|| £ r} are Borelians so they are Lebesgue measurable and thus, estimators like the knn (2) are well defined for arbitrary norms. It is possible, in principle, to consider norms that are not coming from inner products but in this paper we look only at Hilbert norms ||·||_A of the form,

||z||A2 = zTATAz

(57)

where A Î L with L defined as the open set of real non-singular p×p matrices. For each A Î L define the unit sphere,

SA = {x Î Âp: xTATAx £ 1}

(58)

its volume,

bA = l(SA) = ó õ SA  l(dx)

(59)

and the A-symmetric (i.e. ||·||_A radially symmetric) kernel, K_A,

KA(x) = (K°A)(x) = K(Ax)

(60)

where K satisfies assumption 4 and it is I-symmetric, i.e., radially symmetric in the euclidian norm. This means that K(y) depends on y only through the euclidian length of y, i.e. there exists a function F such that,

K(y) = F(yTy)

(61)

The following simple theorem shows that all A-symmetric functions are really of the form (60).

Theorem 6 For any A Î L, [K\tilde] is A-symmetric if and only if we can write

~ K   (x) = K(Ax)  for all  x Î Âp

(62)

for some I-symmetric K.

Proof: [K\tilde](x) is A-symmetric iff [K\tilde](x) = F(||x||_A²) for some function F. Choose K(x) = [K\tilde](A^-1x). This K is I-symmetric since K(x) = F((AA^-1x)^T(AA^-1x)) = F(x^Tx). More over, [K\tilde](x) = [K\tilde](A^-1(Ax)) = K(Ax). Thus, (62) is necessary for A-symmetry. It is also obviously sufficient since the assumed I-symmetry of K in (62) implies that [K\tilde](x) = F((Ax)^T(Ax)) = F(||x||_A²) so it is A-symmetric.
Q.E.D.

An important corollary of theorem 5 is,

Theorem 7 Let A,B Î L. Then, [K\tilde] is AB-symmetric if and only if [K\tilde]_B^-1 is A-symmetric.

Proof: By the first part of theorem 5 we have that [K\tilde] = K°A°B with K some I-symmetric. Thus, [K\tilde]°B^-1 = K°A is A-symmetric by the second part of theorem 5.
Q.E.D.

Let us denote by b(A,K) the total volume that a kernel K assigns to the unit A-sphere S_A, i.e.,

b(A,K) = ó õ SA  K(x) dx

(63)

In order to understand the effect of changing the metric on a density estimator like the kernel (6), it is convenient to write g_n explicitly as a function of everything it depends on; The point z, the metric A, the A-symmetric kernel function [K\tilde], the positive smoothness parameter m and, the data set {x₁,¼,x_n}. Hence, we define,

gn(z;A, ~ K   ,m,{x1,¼,xn}) = 1 n n å j = 1  ~ K   æ ç è xj-z m ö ÷ ø b(A, ~ K   ) mp

(64)

The following result shows that kernel estimation with metric AB is equivalent to estimation of a transformed problem with metric A. The explicit form of the transformed problem indicates that a perturbation of the metric should be regarded as composed of two parts: Shape and volume. The shape is confounded with the form of the kernel and the change of volume is equivalent to a change of the smoothness parameter.

Theorem 8 Let A,B Î L, m > 0, and [K\tilde] an AB-symmetric kernel. Then, for all z Î Â^p and all data sets {x₁¼,x_n} we have,

gn(z;AB, ~ K   ,m,{x1,¼,xn}) = gn( ^ B   z;A, ~ K   °B-1,|B|-1/pm, { ^ B   x1,¼, ^ B   xn})

(65)

where |B| denotes the determinant of B and [^B] = |B|^-1/pB is the matrix B re-scaled to have unit determinant.

Proof: To simplify the notation let us denote,

mB = m |B|1/p

(66)

Substituting AB for A in (64) and using theorem 5 we can write the left hand side of (65) as,

1 n n å j = 1  K æ ç è AB æ ç è xj-z m ö ÷ ø ö ÷ ø b(AB, ~ K   ) mp = 1 n n å j = 1  (K°A) æ ç ç ç è ^ B   xj- ^ B   z mB ö ÷ ÷ ÷ ø b(A,K°A) (mB)p

where K is some I-symmetric kernel and we have made the change of variables x = B^-1y in b(AB,[K\tilde]) (see (63) ). The last expression is the right hand side of (65). Notice that, K°A = [K\tilde]_B^-1 is A-symmetric.
Q.E.D.

5.1  Generalized knn Case with Uniform Kernel

In this section we find the norm of type (57) that minimizes (47) for the estimators of the knn type with uniform kernel which include the double smoothers with K(x) = 1. As it is shown in theorem 5 a change in the determinant of the matrix that defines the norm is equivalent to changing the smoothness parameter. The quantity (57) to be minimized was obtained by fixing the value of the smoothness parameter to the best possible, i.e. the one that minimizes the expression of the (AMSE) (43). Thus, to search for the best norm at a fix value of the smoothness parameter we need to keep the determinant of the matrix that defines the norm constant. We have,

Theorem 9 Consider the problem,

min |A| = 1  æ è ó õ SA  xTH(z)x dx ö ø 2

(67)

where the minimum is taken over all p×p matrices with determinant one, S_A is the unit A-ball and H(z) is the Hessian of the density f Î C² at z which is assumed to be nonsingular.

When H(z) is indefinite, i.e. when H(z) has both positive and negative eigenvalues, the objective function in (67) achieves its absolute minimum value of zero when A is taken as,

A = c-1 diag(   æ  ú Ö x1 p-m   ,¼,   æ  ú Ö xm p-m   ,   æ  ú Ö xm+1 m   ,¼,   æ  ú Ö xp m   ) M

(68)

where the x_j are the absolute value of the eigenvalues of H(z), m is the number of positive eigenvalues, M is the matrix of eigenvectors and c is a normalization constant to get detA = 1 (see the proof for more detailed definitions).

If H(z) is definite, i.e. when H(z) is either positive or negative definite, then for all p×p real matrices A with detA = 1 we have,

ê ê ó õ SA  xTH(z)x dx ê ê ³ 2pp p+3  p ê ê det H(z) ê ê 1/p

(69)

with equality if and only if,

A = H+1/2(z) |H+1/2(z)|1/p

(70)

where H₊^1/2(z) denotes the positive definite square-root of H(z) (see the proof below for explicit definitions).

Proof: Since f Î C² the Hessian is a real symmetric matrix and we can therefore find an orthogonal matrix M that diagonalizes H(z), i.e. such that,

H(z) = MT D M   with    MTM = I

(71)

where,

D = diag(x1,x2,¼,xm,-xm+1,¼, -xp)

(72)

where all the x_j > 0 and we have assumed that the columns of M have been ordered so that all the m positive eigenvalues appear first and all the negative eigenvalues -x_m+1,¼,-x_p appear last so that (71) agrees with (72). Split D as,

D = diag(x1,¼,xm,0,¼,0) - diag(0,¼,0,xm+1,¼,xp) = D+ - D- (73)

and use (71) and (73) to write,

H(z) = MTD+M - MTD-M = (D+1/2M)T(D+1/2M) - (D-1/2M)T(D-1/2M) = S+TS+ - S-TS- (74)

Using (74) and the substitution y = Ax we obtain,

ó õ SA  xTH(z)x dx = ó õ yTy £ 1  yT(A-1)T (S+TS+ - S-TS- ) A-1y dy = ó õ yTy £ 1  < SA-1y, SA-1y > dy (75)

where,

S = S+ + S- = (D+ + D-)1/2 M

(76)

and < .,. > denotes the pseudo-Riemannian inner product,

< x,y > = m å i = 1  xiyi - p å i = m+1  xiyi

(77)

By letting G = diag(1,¼,1,-1,¼,-1) (i.e. m ones followed by p-m negative ones) be the metric with the signature of H(z) we can also write (77) as,

< x,y > = xT G y

(78)

Let,

B = [b1|b2|¼|bp] = SA-1

(79)

where b₁,¼,b_p denote the columns of B. Substituting (79) into (75), using the linearity of the inner product and the symmetry of the unit euclidian sphere we obtain,

ó õ SA  xTH(z)x dx = ó õ yTy £ 1  < By,By > dy = å j  å k  < bj,bk > ó õ SI  yjyk dy (80) = å j  å k  < bj,bk > djk r = r p å j = 1  < bj,bj > (81)

where r stands for,

r = ó õ SI  (y1)2 dy = 2pp p+3

(82)

From (79) and (81) it follows that problem (67) is equivalent to,

min |B| = |S|  æ è p å j = 1  < bj,bj > ö ø 2

(83)

When H(z) is indefinite, i.e. when m Ï {0,p} it is possible to choose the columns of B so that å_j < b_j,b_j > = 0 achieving the global minimum. There are many possible choices but the simplest one is,

B = c · diag(   ___ Öp-m   ,   ___ Öp-m   ,¼,   ___ Öp-m   m  , Öm,Öm,¼,Öm p-m  )

(84)

since,

p å j = 1  < bj,bj > = c2 m (   ___ Öp-m   )2 - c2 (p-m) (Öm)2 = 0.

(85)

The scalar constant c needs to be adjusted to satisfy the constraint that detB = detS. From (79), (84) and (76) we obtain that the matrix for the optimal metric can be written as,

A = B-1S = c-1   ___ Öp-m S+ + c-1 Öm S-

(86)

From (86) we get,

ATA = c-2 p-m S+TS+ + c-2 m S-TS-

(87)

Finally from (74) we can rewrite (87) as,

ATA = c-2 MT æ ç è 1 p-m D+ + 1 m D- ö ÷ ø M (88) = c-2 MT diag( x1 p-m ,¼, xm p-m , xm+1 m ,¼, xp m ) M (89)

Notice that when p-m = m (i.e. when the number of positive equals the number of negative eigenvalues of H(z)) the factor 1/m can be factorized out from the diagonal matrix in (89) and in this case the optimal A can be expressed as,

A = H+1/2(z) |H+1/2(z)|1/p

(90)

where we have used the positive square-root of H(z) defined as,

H+1/2(z) = diag( Ö   x1   ,¼, Ö   xp   )M

(91)

In all the other cases for which H(z) is indefinite, i.e. when m Ï {0,p/2,p} we have,

A = c-1 diag(   æ  ú Ö x1 p-m   ,¼,   æ  ú Ö xm p-m   ,   æ  ú Ö xm+1 m   ,¼,   æ  ú Ö xp m   ) M

(92)

The normalization constant c is fixed by the constraint that detA = 1 as,

c = (p-m)-[m/2p] m-[((p-m))/2p] | det H(z)|[1/2p]

(93)

This shows (68).

Let us now consider the only other remaining case when H(z) is definite, i.e. either positive definite (m = p) or negative definite (m = 0). Introducing l₀ as the Lagrange multiplier associated to the constraint detB = detS we obtain that the problem to be solved is,

min b1,¼bp,l0  L(b1,b2,¼,bp,l0)

(94)

where the Lagrangian L is written as a function of the columns of B as,

L(b1,b2,¼,bp,l0) = æ è p å j = 1  < bj,bj > ö ø 2   - 4 l0 ( det (b1,¼,bp)- det S)

(95)

The -4l₀ instead of just l₀ is chosen to simplify the optimality equations below. The optimality conditions are,

¶L ¶bj = 0   for   j = 1,¼,p   and    ¶L ¶l0 = 0

(96)

where the functional partial derivatives are taken in the Fréchet sense with respect to the column vectors b_j. The Fréchet derivatives of quadratic and multi linear forms are standard text-book exercises. Writing the derivatives as linear functions of the vector parameter h we have,

¶  ¶bj < bj,bj > (h) = 2 < bj,h > (97) ¶  ¶bj det (b1,¼,bp)(h) = det (b1,¼, h j-th col.  ,¼,bp) (98)

Thus, using (97) and (98) to compute the derivative of (95) we obtain that for all h and all j = 1,¼,p we must have,

¶L ¶bj (h) = 2 ì í î p å k = 1  < bk,bk > ü ý þ 2 < bj,h > - 4l0 det (b1,¼,h,¼,bp) = 0

(99)

When å_k < b_k,b_k > ¹ 0 we can rewrite (99) as,

< bj,h > = c0-1 det (b1,¼,h,¼,bp)

(100)

But now we can substitute h = b_i with i ¹ j into (100) and use the fact that the determinant of a matrix with two equal columns is zero, to obtain,

< bj,bi > = 0   for all   i ¹ j.

(101)

In a similar way, replacing h = b_j into (100), we get

< bj,bj > = c0-1 det B = c

(102)

where c is a constant that needs to be fixed in order to satisfy the constraint that detB = detS. We have shown that the optimal matrix B must have orthogonal columns of the same length for the G-metric. This can be expressed with a single matrix equation as,

BTG B = c I

(103)

Substituting (79) into (103) and re-arranging terms we obtain,

ATA = c-1 ST G S (104) = c-1 (S+T + S-T) G (S+ + S-) = c-1 (S+TS+ - S-TS-) ATA = c-1 H(z) (105)

From (105), (103), (82) and (81) we obtain,

ê ê ó õ SA  xT H(z) x  dx ê ê ³ rp |c|

(106)

and replacing the values of r and c we obtain (69).
Q.E.D.

5.2  Yet Another Proof When The Hessian is Definite

Consider the following lemma.

Lemma 1 Let A,B be two p×p non-singular matrices with the same determinant. Then

ó õ SA  ||x||2B dx ³ ó õ SB  ||y||2B dy

(107)

Proof: Just split S_A and S_B as,

SA = (SASB) È(SAScB) (108) SB = (SBSA) È(SBScA) (109)

and write,

ó õ SA  ||x||2B dx = ó õ SB  ||x||2B dx - ó õ ScASB  ||x||2B dx +