Probabilistic Dual Frames and Minimization of Dual Frame Potentials

A sequence of vectors $\{{\bf f}_{i}\}_{i=1}^{\infty}$ in a separable Hilbert space $\mathcal{H}$ is called a frame if there exist $0<A\leq B<\infty$ such that for any ${\bf f}\in\mathcal{H}$,

Furthermore, $\{{\bf f}_{i}\}_{i=1}^{\infty}$ is called a tight frame if $A=B$ and Parseval if $A=B=1$. Frames were firstly introduced by Duffin and Schaeffer in the context of nonharmonic analysis [10] and have been applied in many areas, such as the Kadison-Singer problem [4], time-frequency analysis[15], and wavelet analysis[9]. As the extension of orthonormal basis, a frame permits linear dependence between frame elements and allows each vector in $\mathcal{H}$ to be written as a linear combination of frame elements in a redundant way. A sequence of vectors $\{{\bf g}_{i}\}_{i=1}^{\infty}$ in $\mathcal{H}$ is a dual frame for the frame $\{{\bf f}_{i}\}_{i=1}^{\infty}$ if for any ${\bf f}\in\mathcal{H}$,

An example of dual frames is the canonical dual $\{{\bf S}^{-1}{\bf f}_{i}\}_{i=1}^{\infty}$ where ${\bf S}:\mathcal{H}\rightarrow\mathcal{H}$ is the frame operator and is given by ${\bf S}({\bf f})=\sum_{i=1}^{\infty}\left\langle{\bf f},{\bf f}_{i}\right\rangle{\bf f}_{i},\ \forall\ {\bf f}\in\mathcal{H}$. Indeed, ${\bf S}:\mathcal{H}\rightarrow\mathcal{H}$ is bounded, positive, and invertible. See [7] for more details on frames.

Letting $\mu_{f}:=\sum\limits_{i=1}^{N}\frac{1}{N}\delta_{{\bf y}_{i}}$ be the uniform distribution on $\{{\bf y}_{i}\}_{i=1}^{N}$ where $N\geq n$, $\{{\bf y}_{i}\}_{i=1}^{N}$ being a (finite) frame in the Euclidean space $\mathbb{R}^{n}$ becomes for any ${\bf x}\in\mathbb{R}^{n}$,

Therefore, finite frames can be taken as discrete probabilistic measures on $\mathbb{R}^{n}$, and inspired by this insight, M. Ehler and K. A. Okoudjou in [11, 13] proposed the concept of probabilistic frame, which is a probability measure on $\mathbb{R}^{n}$ satisfying a frame-like condition. Throughout the paper, we let $\mathcal{P}(\mathbb{R}^{n})$ be the set of Borel probability measures on $\mathbb{R}^{n}$ and $\mathcal{P}_{2}(\mathbb{R}^{n})\subset\mathcal{P}(\mathbb{R}^{n})$ be such that

We use $f_{\#}\mu$ to denote the pushforward of $\mu$ by a measurable map $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$:

In particular, ${\bf A}_{\#}\mu$ is used to denote $f_{\#}\mu$ if $f$ is linearly represented by the matrix ${\bf A}$. And the support of $\mu\in\mathcal{P}(\mathbb{R}^{n})$ is given by

where $B_{r}({\bf x})$ is an open ball centered at ${\bf x}$ with radius $r>0$. Then, we have the following definition for probabilistic frames.

Clearly if $\mu\in\mathcal{P}_{2}(\mathbb{R}^{n})$, $\mu$ is Bessel with bound $M_{2}(\mu)$. Furthermore, we define the frame operator ${\bf S}_{\mu}$ for the Bessel probability measure $\mu$ as the following matrix:

Probabilistic frames can be characterized by frame operators as below.

It is noted that probabilistic frames can be studied by optimal transport. Since probabilistic frames are in $\mathcal{P}_{2}(\mathbb{R}^{n})$, then one can use the 2-Wasserstein metric $W_{2}(\mu,\nu)$ to quantify the distance between two probabilistic frames $\mu$ and $\nu$:

Here $\Gamma(\mu,\nu)$ is the set of transport couplings with marginals $\mu$ and $\nu$, i.e.,

where $\pi_{{x}}$, $\pi_{{y}}$ are projections on ${\bf x}$ and ${\bf y}$ coordinates: for any $({\bf x},{\bf y})\in\mathbb{R}^{n}\times\mathbb{R}^{n}$, $\pi_{{x}}({\bf x},{\bf y})={\bf x}$ and $\pi_{{y}}({\bf x},{\bf y})={\bf y}$. For interested readers, we refer to [11, 12, 5, 6] for more details on probabilistic frames, and [14] for optimal transport.

One of our main contributions in this work is about the minimization of probabilistic dual frame potentials. The energy minimization problem is an active topic in mathematics and people in the frame theory community are particularly interested in the minimization of frame potential [2, 12, 8, 1, 18]. Given a unit-norm frame $\{{\bf f}_{i}\}_{i=1}^{N}\subset\mathbb{S}^{n-1}$ where $N\geq d$, its frame potential (FP) is defined as

Benedetto and Fickus in [2] showed that $FP(\{{\bf f}_{i}\}_{i=1}^{N})\geq\frac{N^{2}}{n},$ and the equality holds if and only if the unit-norm frame $\{{\bf f}_{i}\}_{i=1}^{N}$ is tight. Similarly, the authors in [12, 18] defined the probabilistic frame potential (PFP) for a given (Bessel) probability measure $\mu\in\mathcal{P}_{2}(\mathbb{R}^{n})$ and found its lower bound:

and if $\mu\neq\delta_{{\bf 0}}$, the equality holds if and only if $\mu$ is a tight probabilistic frame.

Our work for probabilistic dual frame potentials is motivated by the dual frame potentials for finite frames proposed in [8]. Given a frame $\{{\bf f}_{i}\}_{i=1}^{N}$ and its dual frame $\{{\bf g}_{i}\}_{i=1}^{N}$ in $\mathbb{C}^{n}$, Theorem 2.2 in [8] shows that the dual frame potential (DFP) between $\{{\bf f}_{i}\}_{i=1}^{N}$ and $\{{\bf g}_{i}\}_{i=1}^{N}$ satisfies

and equality holds if and only if $\{{\bf g}_{j}\}_{j=1}^{N}$ is the canonical dual frame $\{{\bf S}^{-1}{\bf f}_{j}\}_{j=1}^{N}$.

The main results given by Theorem 3.5 and Theorem 3.7 in this paper say that given a probabilistic frame $\mu$ (with bounds $A$ and $B$) and its probabilistic dual frame $\nu$, the probabilistic dual frame potential $DPF_{\mu}(\nu)$ between $\mu$ and $\nu$ satisfies

and the equality holds if and only if $\mu$ is a tight probabilistic frame and $\nu$ is the canonical dual ${{\bf S}^{-1}_{\mu}}_{\#}\mu$. And the tightness condition can be dropped: if $T_{\#}\mu$ is a probabilistic dual frame to $\mu$ where the map $T:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ is measurable, then

and the equality holds if and only if $T({\bf y})={\bf S}_{\mu}^{-1}{\bf y}$ for $\mu$ almost all ${\bf y}\in\mathbb{R}^{n}$.

This paper is organized as follows. In Section 2, we show some properties about probabilistic dual frames, and we especially focus on the probabilistic dual frames of pushforward type. In Section 3, we give a lower bound for the probabilistic dual frame potential, which is admitted if and only if the given probabilistic frame is tight and the associated probabilistic dual frame is the canonical dual. And the tightness can be dropped if the dual frame potential is minimized only among probabilistic dual frames of pushforward type. In Section 4, we discuss the generalization of probabilistic dual frames and related open questions. Finally, in Section 5, we give the proof of Proposition 2.3 and Proposition 2.10

This section starts with probabilistic dual frames. Throughout the paper, we use ${\bf A}$ and ${\bf x}$ for a real matrix and vector in $\mathbb{R}^{n}$, and $A$ and $x$ for real numbers. Especially, ${\bf 0}$ is used to denote the zero vector in $\mathbb{R}^{n}$, ${\bf 0}_{n\times n}$ the $n\times n$ zero matrix, and $0$ the real zero number. We use ${\bf Id}$ for the $n\times n$ identity matrix, and ${\bf A}^{t}$ the transpose of matrix ${\bf A}$.

Probabilistic dual frames are also known as transport duals and it was shown that every probabilistic dual frame is a probabilistic frame [17]. ${{\bf S}_{\mu}^{-1}}_{\#}\mu$ is called the canonical (probabilistic) dual frame of $\mu$ (with bounds $\frac{1}{B}$ and $\frac{1}{A}$), since one can let $\gamma:={({\bf Id}\times{\bf S}_{\mu}^{-1})}_{\#}\mu\in\Gamma(\mu,{{\bf S}_{\mu}^{-1}}_{\#}\mu)$. And we have the following equivalence for probabilistic dual frames.

From the above lemma, we know that if $\nu$ is a probabilistic dual frame of $\mu$ with respect to some $\gamma\in\Gamma(\mu,\nu)$, then for any ${\bf f}\in\mathbb{R}^{n}$,

Conversely, we show that the probabilistic dual frame is admitted if the equality holds in a dense subset of $\mathbb{R}^{n}$, and the proof is placed in Section 5.

Recall that given a measurable space $(X,\Sigma,\mu)$, the essential supremum of a measurable function $f:X\rightarrow\mathbb{R}$ with respect to $\mu$, denoted by $\mu$-$\text{esssup}\ f$, is the smallest number $\alpha$ such that the set $\{x:f(x)>\alpha\}$ has measure zero, and if no such number exists, the essential supremum is infinity. By the linearity of matrix trace, we have the following identity that plays a role in obtaining the lower bound of probabilistic dual frame potential.

Lemma 2.4 shows an analogous result from Propitiation 24 in [1], claiming that if $\{{\bf f}_{i}\}_{i=1}^{N}$ is a frame for $\mathbb{R}^{n}$ and $\{{\bf g}_{i}\}_{i=1}^{N}$ is a dual frame of $\{{\bf f}_{i}\}_{i=1}^{N}$, then

and the equality holds if and only if $\langle{\bf f}_{i},{\bf g}_{i}\rangle=\frac{n}{N}$, for each $i$. Indeed, If $\mu=\frac{1}{N}\sum_{i=1}^{N}\delta_{{\bf f}_{i}}$ is a probabilistic frame and $\nu=\frac{1}{N}\sum_{i=1}^{N}\delta_{{\bf h}_{i}}$ a dual frame of $\mu$ with respect to $({\bf Id},T)_{\#}\mu\in\Gamma(\mu,\nu)$ where $T:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ is such that for each $i$, $T({\bf f}_{i})={\bf h}_{i}$, that is to say, $T_{\#}\mu=\nu$. Then by Lemma 2.4,

and the equality holds if and only if $\langle{\bf f}_{i},{\bf h}_{i}\rangle=n$, for each $i$. For the remaining part of this section, we will focus on a particular type of probabilistic dual frame, which are pushforwards of measurable maps.

Indeed, $T_{\#}\mu$ is a dual frame of $\mu$ under $\gamma_{T}:=({\bf Id},T)_{\#}\mu\in\Gamma(\mu,T_{\#}\mu)$, since

Clearly, the canonical probabilistic dual frame ${{\bf S}_{\mu}^{-1}}_{\#}\mu$ is of pushforward type. In [17], Wickman showed that given a probabilistic frame $\mu$, ${\psi_{h}}_{\#}\mu$ is its probabilistic dual frame, where $h:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ is such that $h_{\#}\mu\in\mathcal{P}_{2}(\mathbb{R}^{n})$ and $\psi_{h}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ is

Indeed, the authors in [6] showed that all probabilistic dual frames of pushforward type can be characterized by Wickman’s construction. Note that not every probabilistic dual frame is of pushforward type. For example, $\nu=\frac{1}{2}\delta_{\frac{1}{2}}+\frac{1}{2}\delta_{\frac{3}{2}}$ is a dual frame of $\delta_{1}$ under the coupling $\gamma=(Id,T)_{\#}\nu\in\Gamma(\mu,\delta_{1})$ where $T(x)=1,x\in\mathbb{R}$. However, there does not exist a map $M:\mathbb{R}\rightarrow\mathbb{R}$ such that $\nu=M_{\#}\delta_{1}$.

In the below, we prove that any Bessel probability measure can be transformed into a tight probabilistic frame, and we then show that probabilistic dual frames of pushforward type can be used to characterize tight probabilistic frames.

We have the following trace equality for probabilistic dual frames of pushforward type from Lemma 2.4.

It has been shown that if a probability measure is closed to a given probabilistic frame in some sense, then this probability measure is a frame, and especially, the author gave a sufficient perturbation condition using probabilistic dual frames in Theorem 3.6 of [5]: Suppose $\mu$ is a probabilistic frame and $\nu$ a probabilistic dual frame of $\mu$ with respect to $\gamma_{12}\in\Gamma(\mu,\nu)$. Let $\eta\in\mathcal{P}_{2}(\mathbb{R}^{n})$ and $\gamma_{23}\in\Gamma(\nu,\eta)$. Then, by Gluing Lemma[14, pp.59], there exists $\tilde{\pi}\in\mathcal{P}(\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}^{n})$ with marginals $\gamma_{12}$ and $\gamma_{23}$, and if

then $\eta$ is a probabilistic frame with bounds $\frac{(1-\sigma)^{2}}{M_{2}(\nu)}\ \text{and}\ M_{2}(\eta)$. If the dual frame $\nu$ is of pushforward type, i.e., $\nu=T_{\#}\mu$ for map $T:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$. Then $\gamma_{12}\in\Gamma(\mu,T_{\#}\mu)$ and Gluing Lemma implies $\tilde{\pi}=\gamma_{12}\otimes\eta$. Then the above condition becomes

Indeed, as stated below, the conclusion of $\eta$ being a probabilistic frame holds for any coupling $\gamma\in(\mu,\eta)$ besides the product measure $\mu\otimes\eta$, and its proof is placed at Section 5. The following also generalizes Proposition 3.9 in [5] where $T_{\#}\mu={{\bf S}_{\mu}^{-1}}_{\#}\mu$.

In this section, we study probabilistic dual frame potentials, which is motivated by dual frame potentials for finite frames proposed by [8]. The authors in [8] found a lower bound for dual frame potentials and the minimizer is just the canonical dual. We sketch their statement and proof as below to make the paper complete.

Similarly, we define the probabilistic dual frame potential for a probabilistic dual pair and show that it is invariant under unitary operation.