Spectrally Optimal Dual Frames

• Spectrally optimal dual frames are dual systems that minimize reconstruction error by optimizing the spectrum of frame operators through submajorization and water-filling techniques.
• They balance design constraints by minimizing the spectral radius of erasure error operators, with the canonical dual often serving as the reference under prescribed norm budgets.
• The uniqueness and optimality of these dual frames depend on the frame geometry and connectivity, especially in graph-generated frames where connectedness guarantees a unique spectrally optimal solution.

Spectrally optimal dual frames are dual systems chosen so that reconstruction is extremal with respect to a spectral criterion. The literature considered here studies two closely related optimization problems: one minimizes the spectrum of the dual frame operator under submajorization and convex-potential constraints, and the other minimizes the spectral radius of erasure error operators in robust reconstruction. In both settings, the canonical dual $F^\#=\{S_F^{-1}f_i\}$ is the reference dual, but its optimality and uniqueness depend on the ambient constraints, the geometry of the frame, and, for graph-generated frames, the connectedness of the underlying graph (Massey et al., 2011, Massey et al., 2014, Arati et al., 2024, Deepshikha et al., 2024).

1. Finite-frame setting and canonical duality

Let $H$ be a finite-dimensional Hilbert space and $F=\{f_i\}_{i=1}^N\subset H$ a frame. Its analysis operator is

$T_F(f)=(\langle f,f_i\rangle)_{i=1}^N,$

its synthesis operator is $T_F^*$, and its frame operator is

$$S_F=T_F^*T_F,\qquad S_F(f)=\sum_{i=1}^N \langle f,f_i\rangle f_i.$$

Since $F$ is a frame, $S_F$ is positive and invertible. A sequence $G=\{g_i\}_{i=1}^N$ is a dual of $F$ if

$H$0

The canonical dual is

$H$1

A general dual can be written as

$H$2

where the perturbation satisfies

$H$3

(Krahmer et al., 2012, Arati et al., 2024).

This parametrization is the starting point for both spectral-design problems. In one direction, one studies the possible spectra of frame operators $H$4 as $H$5 ranges over all duals. In the other, one studies erasure error operators built from the pair $H$6 and asks which dual minimizes a worst-case or average spectral-radius criterion.

2. Submajorization, convex potentials, and water-filling constructions

One major line of work defines spectral optimality through the eigenvalues of $H$7. For a fixed frame $H$8, the class

$H$9

admits a precise additive model: every $F=\{f_i\}_{i=1}^N\subset H$0 can be written

$F=\{f_i\}_{i=1}^N\subset H$1

The optimization problem is then to find $F=\{f_i\}_{i=1}^N\subset H$2 such that $F=\{f_i\}_{i=1}^N\subset H$3 is minimal in submajorization among $F=\{f_i\}_{i=1}^N\subset H$4. The solution has a water-filling form: the smallest eigenvalues of $F=\{f_i\}_{i=1}^N\subset H$5 are pushed up to a plateau level $F=\{f_i\}_{i=1}^N\subset H$6, determined by the trace constraint, while the larger eigenvalues remain fixed (Massey et al., 2011).

A related perturbative model imposes both a trace constraint and a proximity constraint to the canonical dual. For

$F=\{f_i\}_{i=1}^N\subset H$7

the frame operators satisfy

$F=\{f_i\}_{i=1}^N\subset H$8

with positivity, trace, operator-norm, and rank restrictions on $F=\{f_i\}_{i=1}^N\subset H$9. The optimal eigenvalue vector is again obtained by a truncated water-filling construction, and the resulting dual simultaneously minimizes every convex potential

$T_F(f)=(\langle f,f_i\rangle)_{i=1}^N,$0

for increasing convex $T_F(f)=(\langle f,f_i\rangle)_{i=1}^N,$1. The frame potential $T_F(f)=(\langle f,f_i\rangle)_{i=1}^N,$2 and the mean squared error $T_F(f)=(\langle f,f_i\rangle)_{i=1}^N,$3 are prominent special cases (Massey et al., 2014).

The same spectral principle extends to shift-generated oblique duals in finitely generated shift-invariant spaces. There the frame operators of all shift-generated $T_F(f)=(\langle f,f_i\rangle)_{i=1}^N,$4-duals form an affine model

$T_F(f)=(\langle f,f_i\rangle)_{i=1}^N,$5

where $T_F(f)=(\langle f,f_i\rangle)_{i=1}^N,$6. Fiberwise Fan–Pall interlacing describes exactly which measurable eigenvalue fields occur, and a non-commutative water-filling operator $T_F(f)=(\langle f,f_i\rangle)_{i=1}^N,$7, with fiber eigenvalues

$T_F(f)=(\langle f,f_i\rangle)_{i=1}^N,$8

produces a unique optimal shift-generated dual $T_F(f)=(\langle f,f_i\rangle)_{i=1}^N,$9 minimizing every non-decreasing convex potential under the prescribed norm budget $T_F^*$0 (Benac et al., 2015).

3. Admissible spectra, tight duals, and conditioning

A complementary spectral question asks which singular-value patterns are possible for dual synthesis operators. Let $T_F^*$1 be the synthesis operator of a fixed frame, with singular values

$T_F^*$2

and let $T_F^*$3, where $T_F^*$4 is the frame length and $T_F^*$5 the ambient dimension. A sequence

$T_F^*$6

is the singular-value list of some dual frame if and only if it satisfies the interlacing inequalities

$T_F^*$7

and

$T_F^*$8

Equivalently, the eigenvalues of $T_F^*$9 are constrained relative to those of the canonical dual (Krahmer et al., 2012).

In this model, the canonical dual corresponds to the minimal admissible spectral profile: $$S_F=T_F^*T_F,\qquad S_F(f)=\sum_{i=1}^N \langle f,f_i\rangle f_i.$$0 Other duals are obtained by adding free columns in an SVD-based parametrization, and this makes it possible to prescribe extremal spectra explicitly. In particular, tight duals are characterized completely. If $$S_F=T_F^*T_F,\qquad S_F(f)=\sum_{i=1}^N \langle f,f_i\rangle f_i.$$1, then for any

$$S_F=T_F^*T_F,\qquad S_F(f)=\sum_{i=1}^N \langle f,f_i\rangle f_i.$$2

there exists a $$S_F=T_F^*T_F,\qquad S_F(f)=\sum_{i=1}^N \langle f,f_i\rangle f_i.$$3-tight dual. If $$S_F=T_F^*T_F,\qquad S_F(f)=\sum_{i=1}^N \langle f,f_i\rangle f_i.$$4, tight duals exist only when the $$S_F=T_F^*T_F,\qquad S_F(f)=\sum_{i=1}^N \langle f,f_i\rangle f_i.$$5 smallest singular values of $$S_F=T_F^*T_F,\qquad S_F(f)=\sum_{i=1}^N \langle f,f_i\rangle f_i.$$6 coincide; when $$S_F=T_F^*T_F,\qquad S_F(f)=\sum_{i=1}^N \langle f,f_i\rangle f_i.$$7, there is exactly one choice $$S_F=T_F^*T_F,\qquad S_F(f)=\sum_{i=1}^N \langle f,f_i\rangle f_i.$$8 (Krahmer et al., 2012).

This spectral design viewpoint makes clear that the canonical dual is not generally optimal for conditioning. Tight duals have condition number $$S_F=T_F^*T_F,\qquad S_F(f)=\sum_{i=1}^N \langle f,f_i\rangle f_i.$$9, whereas the canonical dual may exhibit a larger singular-value spread when the original frame is ill-conditioned. A plausible implication is that “spectrally optimal” must always be interpreted relative to the chosen criterion: minimal singular-value spread, minimal convex potential, and minimal erasure error are distinct objectives, even though they often share the canonical dual in special regimes.

4. Erasure error operators and spectral-radius optimality

A second major line of work defines spectral optimality through robustness to coefficient losses. For a frame $F$0, a dual $F$1, and a diagonal erasure mask $F$2 with exactly $F$3 ones, the error operator is

$F$4

and its spectral radius is

$F$5

The corresponding average error functional is

$F$6

For one erasure,

$F$7

If $F$8 is a 1-uniform frame, meaning $F$9 is constant in $S_F$0, then a dual $S_F$1 is 1-erasure spectrally optimal if and only if

$S_F$2

so the canonical dual of a uniform tight frame is spectrally optimal for one erasure. A uniqueness result also holds under the hypotheses

$S_F$3

where

$S_F$4

and $S_F$5. Under these conditions the canonical dual is the unique minimizer of the 1-erasure average spectral-radius criterion (Mondal et al., 10 Aug 2025).

When both the frame and its dual are allowed to vary, the terminology becomes more rigidly uniform. One source characterizes one-erasure spectral optimality by 1-uniformity of the dual pair and two-erasure spectral optimality by 2-uniformity. In the latter case the optimality condition requires

$S_F$6

and

$S_F$7

The same work compares three measures of error—Frobenius norm, spectral radius, and numerical radius—and states that for Parseval or tight frames all three notions of one-erasure optimality agree, whereas in general they form strictly nested families: $S_F$8 (Arati et al., 2024).

The cited literature also records a limitation: for $S_F$9, a complete characterization of spectral-radius optimizers for a fixed frame is not generally available, although lower bounds, uniform/Parseval cases, and several explicit constructions are known (Mondal et al., 10 Aug 2025).

5. Graph-generated frames and the role of connectedness

For graph-generated frames, the spectral-radius problem becomes particularly explicit. Let $G=\{g_i\}_{i=1}^N$0 be a simple graph on $G=\{g_i\}_{i=1}^N$1 vertices with Laplacian

$G=\{g_i\}_{i=1}^N$2

Assume $G=\{g_i\}_{i=1}^N$3 has exactly $G=\{g_i\}_{i=1}^N$4 nonzero Laplacian eigenvalues $G=\{g_i\}_{i=1}^N$5, choose an orthonormal eigenbasis

$G=\{g_i\}_{i=1}^N$6

let $G=\{g_i\}_{i=1}^N$7 be the $G=\{g_i\}_{i=1}^N$8 matrix of the first $G=\{g_i\}_{i=1}^N$9 eigenvectors, and set

$F$0

The synthesis operator

$F$1

defines a frame $F$2 for $F$3, with frame operator

$F$4

Its canonical dual is

$F$5

(Deepshikha et al., 2024).

In this setting, the structure of all duals is unusually simple. If the graph has $F$6 connected components, then every dual has the form

$F$7

where $F$8 labels the component containing $F$9, and the componentwise offsets satisfy the usual duality constraint (Deepshikha et al., 2024).

The main outcomes are summarized below.

Graph regime	Canonical dual outcome	Uniqueness
Connected graph	Unique spectrally optimal dual for $H$00-erasure and $H$01-erasures; later extended to all $H$02-erasures	Unique
Disconnected graph	Canonical dual remains spectrally optimal for $H$03-erasure, $H$04-erasures, and later all $H$05-erasures	Non-unique; infinitely many optimal offsets

For a connected graph on $H$06 vertices, the associated $H$07-frame satisfies

$H$08

and for every erased pair $H$09,

$H$10

In both cases the canonical dual is optimal, and any alternate dual preserving the same bound must have zero offset, hence uniqueness (Deepshikha et al., 2024).

For disconnected graphs with component sizes $H$11, the canonical dual still attains

$H$12

but nontrivial componentwise offsets can be inserted without increasing the worst-case value. The result is an infinite family of spectrally optimal duals (Deepshikha et al., 2024).

A later extension treats arbitrary $H$13-erasures. It proves that if $H$14 and $H$15 are unitarily equivalent frames with canonical duals $H$16 and $H$17, then

$H$18

For a connected graph-generated frame, the canonical dual is the unique spectrally optimal dual for every $H$19, and for an erasure set $H$20 of size $H$21,

$H$22

when $H$23. For disconnected graphs, the canonical dual remains spectrally optimal for all $H$24, but uniqueness again fails (Deepshikha et al., 27 Jul 2025).

Examples in the literature include the path $H$25, for which $H$26 and $H$27, and the disconnected graph $H$28, for which $H$29 and the family of 1-SODs is infinite (Deepshikha et al., 2024).

6. Weighted models, averaged criteria, and generalized dualities

Weighted one-erasure models replace the unweighted quantity $H$30 by

$H$31

For a fixed frame $H$32, the spectral-radius error of a dual $H$33 becomes

$H$34

and the optimality condition is exact: a dual $H$35 is spectrally optimal if and only if there exists a constant $H$36 such that

$H$37

The set of all such optimal duals is nonempty, convex, and compact (Arati et al., 2024).

A related probabilistic-averaged criterion combines the operator norm and the spectral radius. For an erasure set $H$38,

$H$39

and

$H$40

The corresponding set of $H$41-erasure PASOD-frames is convex, closed, compact, and unitary invariant. In the $H$42 case,

$H$43

For tight frames, the canonical dual is POD if and only if it is PSOD if and only if it is PASOD; moreover, for $H$44, uniqueness of the canonical PASOD-frame is equivalent to

$H$45

(Mondal, 2022).

The framework also extends from ordinary frames to $H$46-frames. If $H$47 is a $H$48-frame and $H$49 a $H$50-dual, then for one erasure

$H$51

After decomposing the frame into linearly connected blocks $H$52 spanning subspaces $H$53, one defines the $H$54-redundancy

$H$55

The optimal one-erasure value is

$H$56

and there always exists a $H$57-dual with

$H$58

The same source gives a closed-form optimum for two erasures in the class of 2-uniform $H$59-duals (Mondal et al., 4 Aug 2025).

A recurring source of confusion is the status of the canonical dual. The results above do not support the claim that it is universally the unique spectrally optimal dual. It is uniquely optimal in several rigid settings—such as connected graph-generated frames for $H$60-erasures and some 1-erasure problems under geometric separation hypotheses—but non-uniqueness appears in disconnected graph models, in weighted Chebyshev-type problems, and in several redundant settings (Deepshikha et al., 27 Jul 2025, Arati et al., 2024, Mondal et al., 10 Aug 2025). Another recurring distinction is between spectral-radius optimality and other error criteria: coincidence holds in Parseval or tight regimes in several papers, but divergence outside those regimes is explicitly documented (Arati et al., 2024, Mondal, 2022).