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Tensor-Tensor Products, Group Representations, and Semidefinite Programming

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Pith's one-line read The paper characterizes exactly which tubes make the star-M product equivariant under a group representation, and uses that to connect tensor semidefinite programs to invariant semidefinite programs.

desk verdict Solid star_M-PSD theory and a genuinely useful equivariant-tube characterization, but the advertised invariant-SDP bridge conflates the tensor action with the congruence action and Example 4.3 fails as written. read the letter →

arxiv 2507.12729 v1 pith:ARKSBT7O submitted 2025-07-17 math.OC cs.CVcs.NAmath.NAmath.RT

Alex Dunbar , Elizabeth Newman This is my paper
classification math.OCcs.CVcs.NAmath.NAmath.RT MSC 15A6990C2220C15
keywords star-MtensorproductsemidefiniteprogrammingpositivetensorsgroupequivarianceSchur'sLemmainvariantSDPcompletionsumsofsquares
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper develops a theory of positive semidefinite third-order tensors in the star-M-product framework and introduces the corresponding semidefinite programs, called M-SDPs. Its central move is to interpret the matrix M in the star-M product as a change of basis adapted to the irreducible representations of an underlying finite group. By doing so, it gives a complete characterization of the tubes whose left-multiplication maps are equivariant under that group action. If the characterization is correct, then M-SDPs with tubes in that space are group-invariant semidefinite programs, which opens the door to block-diagonalized, representation-theoretic computations for tensor completion and sums of squares problems.

What carries the argument

The central object is the matrix representative $M^{{-1}}$ diag(M vec(a)) M of tubal multiplication by a tube a, together with the transform-domain block diagonalization bdiag(A ×_3 M). Equivariance of the multiplication map reduces to the condition that this representative commutes with every ρ(g); once M is chosen as a symmetry-adapted basis, Schur's Lemma forces the diagonal blocks to be scalar multiples of identity on each irreducible component, yielding the explicit subspace W_ρ in Theorem 4.5.

What would settle it

For the S3 example with the permutation representation, take any tube a with a1−a2 ≠ a1−a3 (so a is outside W_ρ) and check whether T_a(g·b) = g·T_a(b) holds for b = e_1 and the transposition g = (12); Theorem 4.5 predicts a nonzero difference unless the constraint a1−a2 = a1−a3 is satisfied, so a single counterexample there would disprove the characterization.

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Extended reading notes

Core claim

For a finite group G with representation ρ and a matrix M representing a change to a symmetry-adapted basis, the tubal multiplication map T_a(b) = a ‹_M b is ρ-equivariant for all b if and only if diag(M vec(a)) = diag(c_1 I_{d_1}, ..., c_m I_{d_m}) for scalars c_j, where the blocks correspond to the irreducible decomposition of $C^{{n_3}}$. This is stated as Theorem 4.5 and gives the complete characterization of equivariant tubes. The same machinery shows that M-PSD tensors are exactly those whose frontal slices in the transform domain are positive semidefinite, so M-SDPs become block-diagonal invariant semidefinite programs when the variable and constraint tubes lie in this equivariant subspace.

Load-bearing premise

The invariant-SDP reading assumes that acting on a tensor along its third mode is the same as conjugating its matrix representative by the group action, and these two actions can disagree for tensors whose tubes are not equivariant.

Editorial extensions

If this is right

  • M-SDPs reduce to n3 independent matrix SDPs when the constraint tensors have only one nonzero frontal slice in the transform domain, which enables parallel computation.
  • M-PSD tensors admit all the classical characterizations of PSD matrices, including a PSD matrix representative, PSD frontal slices in the transform domain, a factorization A = B ‹_M B^J, and square principal minors.
  • The M-nuclear norm has an M-SDP formulation, giving a tensor completion method that the paper demonstrates on video and hyperspectral data with performance that depends on the choice of M.
  • Quadratic forms with Gram tensors whose tubes lie in W_ρ are invariant sums of squares, and the converse holds when each irreducible representation appears with multiplicity at most one.
  • The feasible region of an M-SDP depends on M, as shown by a 2×2×2 example where one orthogonal transformation yields a disk and another yields a line segment.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The characterization suggests a practical preprocessing test for equivariance: given any candidate M and tube a, checking whether diag(M vec(a)) is block-scalar on the irreducible blocks certifies whether the associated linear operator commutes with the group action.
  • For nonabelian groups with faithful representations, full equivariance for all tubes is impossible, so M-SDP can enforce equivariance only on a subspace; users should expect invariant-SDP reductions to be partial rather than complete in such cases.
  • A testable extension is to choose M by an outer optimization loop over the group's irreducibles, using the block-scalar condition as an algebraic constraint, which would make the transformation matrix data-adaptive while preserving equivariance.
  • The invariant-SDP equivalence in Theorem 4.6 should be checked per feasible tensor, because the tensor action g·X = X ×_3 ρ(g) and the congruence action on the matrix representative can disagree for tubes outside W_ρ.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. This paper develops a theory of positive semidefiniteness for third-order tensors equipped with the star_M-product. Section 3 defines M-PSD tensors and proves several equivalent characterizations: a PSD matrix representative, PSD frontal slices in the transform domain, a factorization A = B star_M B^J, square principal minors, and a spectral condition. Section 4 introduces group equivariance for the star_M-product: Theorem 4.3 characterizes when all tubal multiplications are rho-equivariant (M rho(g) M^{-1} diagonal), and Theorem 4.5 characterizes, via Schur's lemma, the subspace W_rho of tubes for which T_a is equivariant. Theorem 4.6 and Example 4.3 aim to connect M-SDPs to invariant SDPs. Sections 5 and 6 give applications to sums-of-squares polynomials and low-rank tensor completion, with numerical experiments on video and hyperspectral data.

Significance. The equivariance characterization in Theorems 4.3 and 4.5 is a solid and elegant contribution: it gives a concrete, checkable description of the tubes that make the star_M-product equivariant, and the proof via Schur's lemma is clean. The M-PSD equivalences in Section 3 carefully extend the t-product theory and are mostly correct. The paper also ships reproducible code, and the tensor-completion experiments demonstrate a genuine computational advantage of decomposing an M-SDP into small per-slice SDPs. However, the claimed bridge to invariant SDPs in Section 4.3.2 conflates the tensor action with a congruence action on matrix representatives; this is a central advertised contribution and needs repair. If reframed correctly, the representation-theoretic results would be suitable for publication after a major revision.

major comments (2)
  1. [Section 4.3.2, Theorem 4.6, and Example 4.3] The claimed bridge to invariant SDPs conflates two different group actions. Section 4.2 defines the tensor action g dot X = X times_3 rho(g), while the proof of Theorem 4.6 establishes invariance only under the congruence action Y -> hat_rho(g)^J Y hat_rho(g) on the matrix representative hat X. These actions are not compatible under the correspondence X -> hat X: for a single tube a, the block of the representation of g dot X is M^J diag(M rho(g) vec(a)) M, whereas the congruent block is rho(g)^J M^J diag(M vec(a)) M rho(g); these are not equal in general. Consequently PSD_M is not invariant under the tensor action. Concretely, in Example 4.3 take X with diagonal tubes (2,0), (2,0) and off-diagonal tubes (1,1); then X is H-PSD, but X times_3 rho(sigma) has transform-domain second slice diag(-sqrt(2), -sqrt(2)), hence is not H-PSD. Thus Theorem 4.6 does not establish that the M-SDP (65) is invariant under the tensor action used elsewhere in the paper; at best it shows that the equivalent block-diagonal matrix SDP is invariant under a congruence action that acts within each n-by-n block, and for X in U this invariance is pointwise trivial. This is a load-bearing gap for the abstract's claim that the star_M setting is a natural setting for the study of invariant semidefinite programs.
  2. [Example 4.3, paragraph after Eq. (74)] Two statements in this example need correction. First, hat_rho(sigma) = I_2 tensor rho(sigma) does not act as (x_i, y_i) -> (x_{g(i)}, y_{g(i)}); on coordinates (x1, x2, y1, y2) it acts as (x2, x1, y2, y1), i.e., it swaps within each of the two pairs, not between pairs. Second, bdiag(hat X) is not fixed pointwise under congruence by hat_rho(sigma): for a feasible X with hat X_{:,:,1} = [[4 alpha, 2 alpha], [2 alpha, 2 alpha]] and hat X_{:,:,2} = diag(4 alpha, 2 alpha), conjugation by rho(sigma) swaps the diagonal entries of the first block. What is true is set-level invariance of the feasible cone. The stress-test's additional claim that hat_rho(sigma) swaps the two 2-by-2 blocks is not correct; the correct action swaps within each block, which still invalidates the paper's pointwise wording.
minor comments (4)
  1. [Section 5.1.1, Theorem 5.2] In the converse direction of Theorem 5.2, the proof asserts the existence of a positive semidefinite Gram matrix Q satisfying f = xi^J Q xi and (I_m tensor rho(g))^J Q (I_m tensor rho(g)) = Q without justification. This is standard by averaging over the finite group, but the argument should be included or cited, since the theorem is a claimed application of the framework.
  2. [Section 4.3.2] The term 'invariant M-SDP' is never defined. Please define explicitly which group action on which variable space is meant before Theorem 4.6; this is closely related to the major comment above and would prevent the current ambiguity.
  3. [Throughout] There are several typos and minor errors: 'Gaterman' should be 'Gatermann'; 'comptaible' should be 'compatible'; 'multiplictaion' should be 'multiplication'; 'strucutre' should be 'structure'. In addition, reference [Bal25] is missing the author's initials and the publisher field is malformed.
  4. [Example 4.2, Eq. (62)] The notation 'a1 := (1,0,0)' and 'a2 := (0,1,1)' is ambiguous because a1 and a2 are tubes, not vectors; write a1 = tube((1,0,0)) and a2 = tube((0,1,1)).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the M-PSD characterizations and equivariant-tube theorem are derived from the stated definitions and Schur's Lemma, with only non-load-bearing citations to prior star_M literature.

full rationale

The paper's central derivation chain is self-contained and non-circular. Definition 3.1 fixes PSD_M by an inner-product inequality, and Propositions 3.2-3.7 prove the slice, factorization, minor, and eigentube characterizations by elementary calculation from that definition; when the star_M-SVD is invoked (Theorem 2.8), it is quoted from [KKA15; KHAN21] as an external, parameter-free theorem with stated assumptions, not assumed as the paper's own conclusion. Theorem 4.5 derives W_rho from the commutation condition M^{-1}diag(M vec(a))M rho(g) = rho(g) M^{-1}diag(M vec(a))M together with Schur's Lemma; W_rho is the proven output, not a pre-defined input, and the proof shows both directions. Theorem 4.6 verifies by direct algebra that for X in U the matrix representative \hat X = (I_n⊗M^J) mat_M(X)(I_n⊗M) is fixed by congruence with \hat rho(g)=I_n⊗rho(g), which is again a calculation rather than a restatement of the conclusion. The M-SOS and nuclear-norm applications use the already-proved PSD characterizations. No fitted parameter is renamed as a prediction; the numerical transforms are fixed or data-dependent experimental choices, not parameters calibrated to the reported completion errors. The invariance statement in Example 4.3 may involve a mismatch between the tensor action X×_3 rho(g) and the congruence action on bdiag(hat X), but that is a correctness concern about the example, not a circular reduction of the paper's theorems to their own inputs. Accordingly, no circular step can be exhibited.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No new entities or fitted constants. The only M-dependence is the choice of transformation matrix, which is an input, not a fitted parameter. The claims rest on standard representation theory and the previously established star_M-product framework.

assumptions (5)
  • domain assumption Properties of the star_M product including the SVD, transpose, and identity tensor are as developed in [KKA15] and [KHAN21].
    Section 2 assumes the star_M-product framework, its SVD (Theorem 2.8), and transpose rules from cited prior work without reproving them.
  • domain assumption M is restricted to orthogonal matrices for the PSD theory and inner product identities.
    Lemma 2.10 and Proposition 3.2 require M in O_{n3}(R); the paper notes orthogonality is necessary.
  • standard math Schur's Lemma over algebraically closed fields and its real-form analogue for totally real representations.
    Used in Theorem 4.5 and Theorem 5.2 to characterize equivariant diagonal maps and invariant blocks.
  • standard math For an invariant SOS polynomial, there exists an invariant Gram matrix via group averaging.
    Invoked in the converse of Theorem 5.2 without proof; standard for finite groups acting orthogonally.
  • domain assumption The decomposition of R^{n3} into irreducibles is totally real and each irreducible appears with multiplicity at most one in the converse of Theorem 5.2.
    Needed for the block-scalar structure of commuting matrices; the paper acknowledges the multiplicity condition and gives a counterexample when it fails.

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Pith. "Pith review of Tensor-Tensor Products, Group Representations, and Semidefinite Programming." pith.science (2026). https://pith.science/paper/ARKSBT7O

@misc{pith2026250712729,
  author       = {Pith},
  title        = {Pith review of: Tensor-Tensor Products, Group Representations, and Semidefinite Programming},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ARKSBT7O}},
  note         = {Machine review of arXiv:2507.12729}
}
abstract

The $\star_M$-family of tensor-tensor products is a framework which generalizes many properties from linear algebra to third order tensors. Here, we investigate positive semidefiniteness and semidefinite programming under the $\star_M$-product. Critical to our investigation is a connection between the choice of matrix M in the $\star_M$-product and the representation theory of an underlying group action. Using this framework, third order tensors equipped with the $\star_M$-product are a natural setting for the study of invariant semidefinite programs. As applications of the M-SDP framework, we provide a characterization of certain nonnegative quadratic forms and solve low-rank tensor completion problems.

Figures

Figures reproduced from arXiv: 2507.12729 by the authors.

Figure 1
Figure 1. Illustration of tensor notation for M P GLn3 pFq. 1.3. Organization. The remainder of the paper is organized as follows. Section 2 fixes notation and provides preliminary results for tensors and tensor-tensor products. In Section 3, we develop a theory of ‹M-PSD tensors and M-semidefinite programs. In Section 4, we prove group equivariance properties for the ‹M￾product and connect ‹M-PSD tensors to invariant SDPs. S… view at source ↗
Figure 2
Figure 2. Illustration of linear constraint tensors for (M-SDP) that lead to n3 indepen￾dent matrix SDPs in the transform domain. For illustration purposes, we consider L “ 5 constraint tensors, each one depicted in a different color. In the transform domain, for each ℓ P rLs, the tensor Aypℓq has only one nonzero frontal slice, resulting in bdiagpAypℓqq having only one nˆn nonzero block. Note that the same frontal slices cou… view at source ↗
Figure 3
Figure 3. The sets tpx, yq | X ľI 0u (left) and tpx, yq | X ľH 0u (right) discussed in Example 3.1. The region for which X ľI 0 is a disk, while the region for which X ľH 0 is a line segment. In this section, we connect (M-SDP) to such invariant semidefinite programs. To do so, we develop an interpretation of the ‹M-product based on the representation theory of finite groups. Specifically, we derive conditions on a finite gro… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Geometric interpretation of ρ-equivariance of symmetric group S3 under the ‹M-product from Example 4.2. The thick black line through the origin depicts V1 “ spanRtp1, 1, 1qu and the blue plane depicts V2 “ spanRtp1, ´1, 0q,p1, 0, ´1qu as the irre￾ducibles of R 3 . The …
Figure 5
Figure 5. Figure 5: Tensor and matrix completion for small video data set Y of size 32 ˆ 32 ˆ 8 [PITH_FULL_IMAGE:figures/full_fig_p026_5.png]
Figure 6
Figure 6. Figure 6: Tensor completion for large video data set Y of size 64 ˆ 64 ˆ 128. 6.2. Hyperspectral Data. We solve the tensor completion problem (97) for the Indian Pines hyperspectral dataset [BBL15], originally downloaded from Matlab’s Hyperspectral Imaging Toolbox5 . Hyperspectr…
Figure 7
Figure 7. Figure 7: Tensor completion for hyperspectral data set Y of size 145 ˆ 145 ˆ 220 [PITH_FULL_IMAGE:figures/full_fig_p029_7.png]

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