Consistency of Probability Measure Quantization by Means of Power Repulsion–Attraction Potentials

Abstract

This paper is concerned with the study of the consistency of a variational method for probability measure quantization, deterministically realized by means of a minimizing principle, balancing power repulsion and attraction potentials. The proof of consistency is based on the construction of a target energy functional whose unique minimizer is actually the given probability measure $\omega$ to be quantized. Then we show that the discrete functionals, defining the discrete quantizers as their minimizers, actually $\Gamma$-converge to the target energy with respect to the narrow topology on the space of probability measures. A key ingredient is the reformulation of the target functional by means of a Fourier representation, which extends the characterization of conditionally positive semi-definite functions from points in generic position to probability measures. As a byproduct of the Fourier representation, we also obtain compactness of sublevels of the target energy in terms of uniform moment bounds, which already found applications in the asymptotic analysis of corresponding gradient flows. To model situations where the given probability is affected by noise, we further consider a modified energy, with the addition of a regularizing total variation term and we investigate again its point mass approximations in terms of $\Gamma$-convergence. We show that such a discrete measure representation of the total variation can be interpreted as an additional nonlinear potential, repulsive at a short range, attractive at a medium range, and at a long range not having effect, promoting a uniform distribution of the point masses.

Appendix: Conditionally Positive Definite Functions

In order to compute the Fourier representation of the energy functional $\mathcal {E}$ in Sect. 3.1.3, we used the notion of generalized Fourier transforms and conditionally positive definite functions from [36], which we shall briefly recall here for the sake of completeness. In fact, the main result reported below, Theorem 7.6 is shown in a slightly modified form with respect to [36, Theorem 8.16], to allow us to prove the moment bound in Sect. 4. The representation formula (3.10) is a consequence of Theorem 7.4 below, which serves as a characterization formula in the theory of conditionally positive definite functions.

Definition 7.1

[36, Definition 8.1] Let $\mathbb {P}_{k}(\mathbb {R}^d)$ denote the set of polynomial functions on $\mathbb {R}^d$ of degree less or equal than k . We call a continuous function $\Phi :\mathbb {R}^d \rightarrow \mathbb {C}$ conditionally positive semi-definite of order m if for all $N\in \mathbb {N}$, pairwise distinct points $x_1,\ldots ,x_N \in \mathbb {R}^d$, and $\alpha \in \mathbb {C}^N$ with

$$\begin{aligned} \sum _{j=1}^{N} \alpha _j p(x_j) = 0, \quad \text {for all } p \in \mathbb {P}_{m - 1}(\mathbb {R}^d), \end{aligned}$$

(7.1)

the quadratic form given by $(\Phi (x_j - x_k))_{jk}$ is non-negative, i.e.,

$$\begin{aligned} \sum _{j,k=1}^{N} \alpha _j \overline{\alpha _k} \Phi (x_j - x_k) \ge 0. \end{aligned}$$

Moreover, we call $\Phi$ conditionally positive definite of order m if the above inequality is strict for $\alpha \ne 0$.

1.1 Generalized Fourier Transform

When working with distributional Fourier transforms, which can serve to characterize the conditionally positive definite functions defined above, it can be opportune to reduce the standard Schwartz space $\mathcal {S}$ to functions which in addition to the polynomial decay for large arguments also exhibit a certain decay for small ones. In this way, one can elegantly neglect singularities in the Fourier transform at 0, which could otherwise arise.

Definition 7.2

(Restricted Schwartz class $\mathcal {S}_m$ ) [36, Definition 8.8] Let $\mathcal {S}$ be the Schwartz space of functions in $C^\infty (\mathbb {R}^d)$ which for $\left| x \right| \rightarrow \infty$ decay faster than any fixed polynomial. Then, for $m \in \mathbb {N}$, we denote by $\mathcal {S}_m$ the subset of those functions $\gamma$ in $\mathcal {S}$ which additionally fulfill

$$\begin{aligned} \gamma (\xi ) = O(\left| \xi \right| ^m) \quad \text {for } \xi \rightarrow 0. \end{aligned}$$

(7.2)

Furthermore, we shall call an (otherwise arbitrary) function $\Phi :\mathbb {R}^d \rightarrow \mathbb {C}$ slowly increasing if there is an $m \in \mathbb {N}$ such that

$$\begin{aligned} \Phi (x) = O \left( \left| x \right| ^m \right) \quad \text {for } \left| x \right| \rightarrow \infty . \end{aligned}$$

Definition 7.3

(Generalized Fourier transform) [36, Definition 8.9] For $\Phi :\mathbb {R}^d \rightarrow \mathbb {C}$ continuous and slowly increasing, we call a measurable function $\widehat{\Phi } \in L_{\mathrm {loc}}^2(\mathbb {R}^d {\setminus } \left\{ 0 \right\} )$ the generalized Fourier transform of $\Phi$ if there exists a multiple of $\frac{1}{2}$, $m = \frac{1}{2}n, \, n \in \mathbb {N}_0$ such that

$$\begin{aligned} \int _{\mathbb {R}^d} \Phi (x)\widehat{\gamma }(x) \mathrm {d}x = \int _{\mathbb {R}^d} \widehat{\Phi }(\xi ) \gamma (\xi ) \mathrm {d}\xi \quad \text {for all } \gamma \in \mathcal {S}_{2m}. \end{aligned}$$

(7.3)

Then, we call m the order of $\widehat{\Phi }$.

Note that the order here is defined in terms of 2m instead of m , which is why we would like to allow for multiples of $\frac{1}{2}$.

1.2 Representation Formula for Conditionally Positive Definite Functions

Theorem 7.4

[36, Corollary 8.13] Let $\Phi :\mathbb {R}^d \rightarrow \mathbb {C}$ be a continuous and slowly increasing function with a non-negative, non-vanishing generalized Fourier transform $\widehat{\Phi }$ of order m that is continuous on $\mathbb {R}^d {\setminus } \left\{ 0 \right\}$. Then, for pairwise distinct points $x_1, \dots , x_N \in \mathbb {R}^d$ and $\alpha \in \mathbb {C}^N$ which fulfill condition (7.1), i.e.,

$$\begin{aligned} {\sum _{j=1}^{N} \alpha _j p(x_j) = 0, \quad \text {for all } p \in \mathbb {P}_{m - 1}(\mathbb {R}^d),} \end{aligned}$$

we have

$$\begin{aligned} \sum _{j,k=1}^N \alpha _j \overline{\alpha }_k \Phi \left( x_j-x_k \right) = \int _{\mathbb {R}^d} \left| \sum _{j=1}^{N} \alpha _j \mathrm {e}^{\mathrm {i}x_j \cdot \xi }\right| ^2 \widehat{\Phi }(\xi ) \mathrm {d}\xi . \end{aligned}$$

(7.4)

1.3 Computation for the Power Function

Given Theorem 7.4, in this paper we are naturally interested in the explicit formula of the generalized Fourier transform for the power function $x \mapsto \left| x \right| ^q$ for $q \in [1,2)$. It is a nice example of how to pass from an ordinary Fourier transform to the generalized Fourier transform by extending the formula by means of complex analysis methods. Our starting point will be the multiquadric $x \mapsto \left( c^2 + \left| x \right| ^2 \right) ^{\beta }$ for $\beta < -d/2$, whose Fourier transform involves the modified Bessel function of the third kind:

For $\nu \in \mathbb {C}$, $z \in \mathbb {C}$ with $\left| {\text {arg}} z \right| < \pi /2$, define

$$\begin{aligned} K_\nu (z) := \int _{0}^\infty \exp (-z \cosh (t)) \cosh (\nu t) \mathrm {d}t, \end{aligned}$$

the modified Bessel function of the third kind of order $\nu \in \mathbb {C}$.

Theorem 7.5

[36, Theorem 6.13] For $c > 0$ and $\beta < -d/2$,

$$\begin{aligned} \Phi (x) = (c^2+\left| x \right| ^2)^{\beta }, \quad x \in \mathbb {R}^d, \end{aligned}$$

has (classical) Fourier transform given by

$$\begin{aligned} \widehat{\Phi }(\xi ) =(2\pi )^{d/2} \frac{2^{1+\beta }}{\Gamma (-\beta )} \left( \frac{\left| \xi \right| }{c} \right) ^{-\beta -d/2}K_{d/2+\beta }(c \left| \xi \right| ). \end{aligned}$$

(7.5)

In the following result, we have slightly changed the statement compared to the original reference [36, Theorem 8.16] in order to allow orders which are a multiple of 1 / 2 instead of just integers. The latter situation made sense in [36] because the definition of the order involves the space $\mathcal {S}_{2m}$ due to its purpose in the representation formula of Theorem 7.4, where a quadratic form appears. However, in Sect. 4 we need the generalized Fourier transform in the context of a linear functional, hence a different range of orders. Fortunately, one can easily generalize the proof in [36] to this fractional case, as all integrability arguments remain true when permitting multiples of 1 / 2 , in particular the estimates in (7.8) and (7.10).

Theorem 7.6

• 1. [36, Theorem 8.15] $\Phi (x) = (c^2 + \left| x \right| ^2)^\beta$, $x \in \mathbb {R}^d$ for $c > 0$ and $\beta \in \mathbb {R} {\setminus } \mathbb {N}_0$ has the generalized Fourier transform

  $$\begin{aligned} \widehat{\Phi }(\xi ) = (2\pi )^{d/2} \frac{2^{1+\beta }}{\Gamma (-\beta )} \left( \frac{\left| \xi \right| }{c} \right) ^{-\beta -d/2} K_{d/2+\beta }(c \left| \xi \right| ), \quad \xi \ne 0 \end{aligned}$$

  (7.6)

  of order $m = \max (0, {\lfloor 2\beta + 1\rfloor /2)}$.

• 2. [36, Theorem 8.16] $\Phi (x) = \left| x \right| ^\beta$, $x \in \mathbb {R}^d$ with $\beta \in \mathbb {R}_+ {\setminus } 2\mathbb {N}$ has the generalized Fourier transform

  $$\begin{aligned} \widehat{\Phi }(\xi ) = (2\pi )^{d/2}\frac{2^{\beta +d/2}\Gamma ((d+\beta )/2)}{\Gamma (-{\beta }/2)} \left| \xi \right| ^{-\beta -d}, \quad \xi \ne 0. \end{aligned}$$

  of order $m = {\lfloor \beta + 1\rfloor /2}$.

Note that in the cases of interest to us, the second statement of the theorem means that the generalized Fourier transform of $\Phi (x) = \left| x \right| ^\beta$ is of order $\frac{1}{2}$ for $\beta \in (0, 1)$ and 1 for $\beta \in [1,2)$, respectively. As this statement appears in a slightly modified form with respect to [36, Theorem 8.16] we report below an explicit, although rather concise proof of it.

Proof

• 1. We can pass from formula (7.5) to (7.6) by analytic continuation, where the exponent m serves to give us the needed integrable dominating function, see formula (7.8) below. Let $G = \left\{ \lambda \in \mathbb {C} : \mathrm{Re}(\lambda ) < m \right\} {\ni \beta }$ and

  $$\begin{aligned} \varphi _\lambda (\xi ) := {}&(2\pi )^{d/2} \frac{2^{1+\lambda }}{\Gamma (-\lambda )} \left( \frac{\left| \xi \right| }{c} \right) ^{-\lambda -d/2} K_{d/2+\lambda }(c \left| \xi \right| )\\ \Phi _\lambda (\xi ) := {}&\left( c^2 + \left| \xi \right| ^2 \right) ^\lambda . \end{aligned}$$

  We want to show that for all $\lambda \in G$

  $$\begin{aligned} \int _{\mathbb {R}^d} \Phi _\lambda (\xi )\widehat{\gamma }(\xi ) \mathrm {d}\xi = \int _{\mathbb {R}^d} \varphi _\lambda (\xi ) \gamma (\xi ) \mathrm {d}\xi , \quad \text {for all } \gamma \in \mathcal {S}_{2m}, \end{aligned}$$

  which is so far true for $\lambda$ real and $\lambda < {-}d/2$ by (7.5). As the integrands $\Phi _\lambda \widehat{\gamma }$ and $\varphi _\lambda \gamma$ are analytic, they can be expressed in terms of Cauchy integral formulas. The integral functions

  $$\begin{aligned} f_1(\lambda )= & {} \int _{\mathbb {R}^d} \Phi _\lambda (\xi )\widehat{\gamma }(\xi ) \mathrm {d}\xi = \int _{\mathbb {R}^d} \frac{1}{2 \pi i} \int _{\mathcal C} \frac{\Phi _z(\xi )}{z - \lambda } dz \widehat{\gamma }(\xi ) d\xi \\ f_2(\lambda )= & {} \int _{\mathbb {R}^d} \varphi _\lambda (\xi ) \gamma (\xi ) \mathrm {d}\xi = \int _{\mathbb {R}^d} \frac{1}{2 \pi i} \int _{\mathcal C} \frac{\varphi _z(\xi )}{z - \lambda } dz {\gamma }(\xi ) d\xi , \end{aligned}$$

  will be also analytic as soon as we can find a uniform dominating function of the integrands on an arbitrary compact curve $\mathcal {C} \subset G$, to allow the application of Fubini-Tonelli’s theorem and derive corresponding Cauchy integral formulas for $f_1$ and $f_2$ (see the details of the proofs of [36, Theorem 8.15] and [36, Theorem 8.16]). A dominating function for the integrand of $f_1(\lambda )$ is easily obtained thanks to the decay of $\widehat{\gamma } \in \mathcal {S}$ faster of any polynomially growing function (notice that $\mathrm{Re}(\lambda )<m$). It remains to find a dominating function for the integrand of $f_2(\lambda )$. Setting $b := \mathrm{Re}(\lambda )$, for $\xi$ close to 0 we get, by using the bound

  $$\begin{aligned} \left| K_\nu (r) \right| \le {\left\{ \begin{array}{ll} 2^{\left| \mathrm{Re}(\nu ) \right| - 1}\Gamma \left( \left| \mathrm{Re}(\nu ) \right| \right) r^{-\left| \mathrm{Re}(\nu ) \right| }, &{}\mathrm{Re}(\nu ) \ne 0,\\ \frac{1}{\mathrm {e}}-\log \frac{r}{2},&{}r < 2, \mathrm{Re}(\nu ) = 0. \end{array}\right. } \end{aligned}$$

  (7.7)

  for $\nu \in \mathbb {C}, r > 0$, as derived in [36, Lemma 5.14], that

  $$\begin{aligned} \left| \varphi _z(\xi ) \gamma (\xi ) \right| \le C_\gamma \frac{2^{b+\left| b + d/2 \right| }\Gamma (\left| b + d/2 \right| )}{\left| \Gamma (-\lambda ) \right| }c^{b+d/2-\left| b+d/2 \right| }\left| \xi \right| ^{-b-d/2-\left| b+d/2 \right| +2m} \end{aligned}$$

  (7.8)

  for $b \ne -d/2$ and

  $$\begin{aligned} \left| \varphi _z(\xi )\gamma (\xi ) \right| \le C_{z} \frac{2^{1-d/2}}{\left| \Gamma (-\lambda ) \right| }\left( \frac{1}{\mathrm {e}} - \log \frac{c \left| \xi \right| }{2} \right) . \end{aligned}$$

  for $b = -d/2$. Taking into account that $\mathcal {C}$ is compact and $1/\Gamma$ is an entire function, this yields

  $$\begin{aligned} \left| \varphi _z(\xi )\gamma (\xi ) \right| \le C_{m,c,\mathcal {C}} \left( 1 + \left| \xi \right| ^{-d+2\varepsilon }-\log \frac{c \left| \xi \right| }{2} \right) , \end{aligned}$$

  with $\left| \xi \right| < \min \left\{ 1/c,1 \right\}$ and $\varepsilon := m-b{>0}$, which is locally integrable. For $\xi$ large, we similarly use the estimate for large r ,

  $$\begin{aligned} \left| K_\nu (r) \right| \le \sqrt{\frac{2\pi }{r}} \mathrm {e}^{-r} \mathrm {e}^{\left| \mathrm{Re}(\mu ) \right| ^2/(2r)}, \quad r > 0, \end{aligned}$$

  (7.9)

  from [36, Lemma 5.14] to obtain

  $$\begin{aligned} \left| \varphi _z(\xi )\gamma (\xi ) \right| \le C_{\mathcal C} \frac{2^{1+b}\sqrt{2\pi }}{\left| \Gamma (-\lambda ) \right| }c^{b+(d-1)/2} \left| \xi \right| ^{-b-(d+1)/2} \mathrm {e}^{-c \left| \xi \right| } \mathrm {e}^{\left| b+d/2 \right| ^2/(2c \left| \xi \right| )} \end{aligned}$$

  and consequently

  $$\begin{aligned} \left| \varphi _\lambda (\xi )\gamma (\xi ) \right| \le C_{\gamma ,m,\mathcal {C},c} \mathrm {e}^{-c \left| \xi \right| }, \end{aligned}$$

  which certainly is integrable.

• 2. We want to pass to $c \rightarrow 0$ in formula (7.6). This can be done by applying the dominated convergence theorem in the definition of the generalized Fourier transform (7.3). Writing $\Psi _c(x) := \left( c^2+ \left| x \right| ^2 \right) ^{\beta /2}$ for $c>0$, we know that

  $$\begin{aligned} \widehat{\Psi }_c(\xi ) = \psi _c(\xi ) := (2\pi )^{d/2} \frac{2^{1+\beta /2}}{\left| \Gamma (-\beta /2) \right| } \left| \xi \right| ^{-\beta -d}(c \left| \xi \right| )^{(\beta +d)/2}K_{(\beta +d)/2}(c \left| \xi \right| ). \end{aligned}$$

  By using the decay properties of a $\gamma \in \mathcal {S}_{2m}$ in the estimate (7.8), we get

  $$\begin{aligned} \left| \psi _c(\xi )\gamma (\xi ) \right| \le C_\gamma \frac{2^{\beta +d/2}\Gamma ((\beta +d)/2}{\left| \Gamma (-\beta /2) \right| } \left| \xi \right| ^{2m-\beta -d} \quad \text {for } \left| \xi \right| \rightarrow 0 \end{aligned}$$

  (7.10)

  and

  $$\begin{aligned} \left| \psi _c(\xi )\gamma (\xi ) \right| \le C_\gamma \frac{2^{\beta +d/2}\Gamma ((\beta +d)/2)}{\left| \Gamma (-\beta /2) \right| } \left| \xi \right| ^{-\beta -d}, \end{aligned}$$

  yielding the desired uniform dominating function. The claim now follows by also taking into account that

  $$\begin{aligned} \lim _{r\rightarrow 0} r^\nu K_\nu (r) = \lim _{r\rightarrow 0} 2^{\nu -1} \int _{0}^{\infty } \mathrm {e}^{-t} \mathrm {e}^{-r^2/(4t)} t^{\nu -1} \mathrm {d}t = 2^{\nu -1} \Gamma (\nu ). \end{aligned}$$

  $\square$