1 Introduction
Let and be two separable Hilbert spaces of dimension and , possibly infinite. Let denote the class of linear bounded operators from to and where denotes the rank operator. In this work, we are interested in characterizing the solutions of the following constrained optimization problem
(1) 
where is some operator norm and some subset to be specified in due time, and where the symbol denotes the operator composition. Problem (1) is nonconvex due to the rank constraint and is in general infinitedimensional. This raises the question of the tractability of problems of the form of (1).
In the last decade, there has been a surge of interest for lowrank solutions of linear matrix equations [3, 6, 8, 9, 11, 14]. Problems of the form of (1) can be viewed as generalizations to the infinite dimensional case of some of those matrix equations. In the finite dimensional case, certain instances with a very special structures admit closedform solutions [2, 4, 10, 12], which can be computed in polynomial time. We mention that some authors have proposed tractable but suboptimal solutions to some particular finite [1, 7, 16, 17] and infinitedimensional [18] problem instances.
In this work, we show that some infinitedimensional problems of the form of (1) admit also a closedform solution. The proof relies on the wellknow SchmidtEckhartYoungMirsky theorem [15]. The theorem exposed in this work can be viewed as a direct generalization to the infinite dimensional case of [4, Theorem 4.1]. It also generalizes the solution of approximation problems in the sense of the th order Schatten norm, and includes the Frobenius norm as a particular case.
2 Problem Statement and Solution
We begin by introducing some notations, then define the lowrank approximation problem and finally provide a closedform solution and error norm.
2.1 Notations
Let and be any ONBs of and . The inner product in those spaces will be denoted by and and their induced norm by and . Let Let
be the identity operator. Let the singular value decomposition (SVD) of
bewhere , are respectively the left and right singular functions associated to the singular values of [19]. The pseudo inverse of denoted will be defined as
2.2 Optimization Problem
We are now ready to clarify the definition of problem (1). Let . We are interested in the lowrank approximation problem solutions
(2) 
where symbol denotes an operator composition.
2.3 ClosedForm Solution
We detail in the following theorem our result. The proof is detailed in Section 4.
Problem (2) admits the optimal solution
where is given by and . Moreover, the square of the approximation error is
(3) 
where .
Remark 1 (Modified th Schatten Norm)
The result can be extended for an approximation in the sense of the modified th Schatten norm. In particular, for and for , this extension can be seen as the DMD counterpart to the POD problem with energy inner product presented in [13, Proposition 6.2]. Let us define this modified norm. We need first to introduce an additional norm for induced by an alternative inner product. For any , we define
where is compact and selfadjoint, i.e., . Since is selfadjoint, the SVD guarantees that can be decomposed as where , and that . The modified th Schatten norm is then defined for any and as
which can be rewritten as
An optimal solution of problem (2) in the sense of the norm is then
where with .
3 Some Particularizations
3.1 Trace Norm, HilbertSchmidt Norm and Induced Norm
For , the th Schatten norm corresponds to the induced norm
while for and we obtain the trace norm and the HilbertSchmidt norm.
We have thus shown that is the optimal solution of problem (2) for approximation in the sense of the trace norm, the HilbertSchmidt norm or the induced norm.
3.2 Unconstrained DMD, Lowrank DMD and KernelBased DMD
If is full rank, or equivalently , then and the optimal approximation error simplifies to
If and , we recover the standard result for the unconstrained DMD problem [16].
In the case and , we recover the optimal result proposed in [4, Theorem 4.1] for lowrank DMD (or extended DMD in finite dimension). Suboptimal solutions to this problem have been proposed in [1, 7, 18, 17]
In the case , the result characterizes the solution of lowrank approximation in reproducing kernel Hilbert spaces, on which kernelbased DMD relies. Theorem 2.3 justifies in this case the solution computed by the optimal kernelbased DMD algorithm proposed in [5]. We note that the proposed solution has been already given in [18] for the infinite dimensional setting, but in the case where . Nevertheless, the solution provided by the authors is suboptimal in the general case.
3.3 Continuous DMD
In the case , the result characterizes the solution of a continuous version of the DMD problem, where the number of snapshots are infinite. In particular, for , the problem is the DMD counterpart to the continuous POD problem presented in [13, Theorem 6.2]. Here, problem (2) is defined as follows. in (2) are compact HilbertSchmidt operators, defined by their kernels
where are the HilbertSchmidt kernels with supplied by the measure and so that . The solution and the optimal error are then characterized by Theorem 2.3.
4 Proof of Theorem 2.3
We will use the following extra notations in the proof.
We define , where for any ,
. We thus have
Finally, let
and
4.1 ClosedForm Solution
We begin by proving that problem (2) admits the solution .
First, we remark that is fullrank () so that , with . Therefore, using the Pythagore Theorem, we have
Since we have
we obtain
(4) 
where the last equality follows from the invariance of the th Schatten norm to unitary transforms and the fact that .
Second, from the Sylvester inequality, we get
and by invariance of the th Schatten norm to unitary transforms, we obtain
where .
Third, from the SchmidtEckhartYoungMirsky theorem [15], problem () admits the solution
Fourth, we remark that and that the truncation to terms of the SVD of corresponds to the operator yielding . Therefore, and we verify that
We deduce that the minimum of the objective function of () reaches the minimum of the objective function of () at , i.e.,
4.2 Characterization of the Optimal Error Norm
It remains to characterize the error norm (3). On the one hand, we have
Since is a ONB of , we can expand the norm and obtain
where, in order to obtain the two last equalities, we have exploited the fact that is an ONB of . On the other hand, we have
Finally, from (4.1) and the two above expressions, we conclude
5 Conclusion
We have shown that there exists a closedform optimal solution to the nonconvex problem related to lowrank approximation of linear bounded operators in the sense of the th Schatten norm. This result generalizes to lowrank operator in Hilbert spaces solutions obtained in the context of lowrank matrix approximation. As in the latter finitedimensional case, the proposed closedform solution takes the form of the orthogonal projection of the solution of the unconstrained problem onto a specific lowdimensional subspace. However, the proof is substantially different. It relies on the wellknown SchmidtEckhartYoungMirsky theorem. The proposed theorem is discussed and applied to various contexts, including lowrank approximation with respect to the trace norm, the induced norm and the HilbertSchmidt norm, or kernelbased and continuous DMD.
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