{"id":"858366b8-1efd-44a9-a7bc-43d8e465ebe0","arxiv_id":"1908.09275","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A single family of Alpha Procrustes metrics unifies the Bures-Wasserstein and Log-Euclidean distances on positive definite matrices and operators, with explicit geodesics and closed forms in reproducing kernel Hilbert spaces.","lead":"A new one-parameter family of distances, the Alpha Procrustes distances, interpolates between two important ways to measure the difference between positive definite matrices and operators. It connects the Bures-Wasserstein distance used in optimal transport with the Log-Euclidean distance used in medical imaging and machine learning, and extends to infinite dimensions.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Infinite-dimensional Alpha Procrustes distances are defined only within fixed-scalar fibers PC2(H)(γ); no cross-γ distance is given, so the claimed unification with Log-Hilbert-Schmidt on all PC2(H) is incomplete.","rationale":"The reader's weakest assumption was Lemma 1, the claim that the unitary polar factor of an invertible unitized Hilbert-Schmidt operator lies in U(H)∩HSX(H). That lemma is actually secure: for A+γI with A∈HS, the modulus is γI+K with K∈HS, so its inverse is (1/γ)I+K' with K'∈HS, and the unitary factor is I+R with R∈HS. The reader also flagged repeated-row blocks in Eqs. (74), (79), and (97) and a different limiting matrix in Corollary 5 as apparent errors. These are consistent with the factorization T = (A h, B h, A h A^*B h)(A^*, B^*, B^*)^T, which forces the second and third block rows to coincide, and the limit in Corollary 5 follows from the γ^{4α} scaling of the block matrix. The finite-dimensional claims are well supported: d^α_proE is a constant multiple of the Bures-Wasserstein distance between A^{2α} and B^{2α}, and the Riemannian submersion construction gives the geodesic formula. The genuine load-bearing weakness is the infinite-dimensional scope: the Alpha Procrustes family is constructed only on each fixed-scalar fiber PC2(H)(γ), not on PC2(H) as a whole, while the Log-Hilbert-Schmidt distance is defined across fibers. This is acknowledged in Remark 2, but it limits the stated unifying contribution. The paper should either extend the construction to γ≠ν or explicitly qualify the abstract and title claims to fixed-scalar subsets and trace-class limits.","tokens_in":30049,"tokens_out":44102,"duration_ms":431265,"concrete_test":"Take H=ℓ2 and consider P=2I, Q=3I, both in PC2(H). Search §3 and §3.2 for any definition or theorem that assigns d^α_proHS(P,Q) a finite value: Definition 3 requires a common γ, and Theorem 11 and Theorem 13 are stated only for PC2(H)(γ). If no formula applies, the claim that the family is defined on PC2(H) fails. As a second check, attempt to compute the α→0 limit of any proposed cross-γ extension and verify whether it equals ||log(2I)−log(3I)||_HSX = |log(2/3)|, as required for unification with the Log-Hilbert-Schmidt distance.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The paper's abstract and Section 3 claim generalization to PC2(H), the full set of positive definite unitized Hilbert-Schmidt operators. However, Definition 3, Theorem 11, and Theorem 13 only define the Alpha Procrustes distance for pairs with the same scalar part, that is (A+γI),(B+γI) ∈ PC2(H)(γ). The fibers PC2(H)(γ) are disjoint: if A+γI = B+νI with γ≠ν, then (γ−ν)I = B−A ∈ HS(H), impossible because a nonzero multiple of I is not Hilbert-Schmidt. Thus no distance is defined between operators with different scalar parts. The Log-Hilbert-Schmidt distance in Eq. (43), by contrast, is defined on all PC2(H) and does handle different scalar parts. Consequently, the claimed recovery of Log-Hilbert-Schmidt as the α→0 limit is only established within a fixed fiber, not on the union PC2(H). Remark 2 explicitly defers the case γ≠ν to future work, which confirms that this is not an oversight in a proof step but a genuine scope gap in the central infinite-dimensional contribution.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a one-parameter family of \"Alpha Procrustes\" distances on the SPD matrix cone, defined by minimizing a Frobenius norm between scaled powers A^α and B^α U over unitaries. It proves an explicit trace formula, shows that α=1/2 gives twice the Bures-Wasserstein distance and that α→0 gives the Log-Euclidean distance, compares the family with power-Euclidean distances, and constructs a Riemannian submersion whose distance is the Alpha Procrustes distance. The construction is then extended to positive definite unitized Hilbert-Schmidt operators of the form A+γI, with the scalar part γ fixed, recovering a fiberwise Bures-Wasserstein distance and a fiberwise Log-Hilbert-Schmidt limit. The last sections provide closed-form RKHS Gram-matrix formulas for covariance operators and corresponding distances between Gaussian measures.","tokens_in":30242,"tokens_out":16679,"duration_ms":160900,"significance":"The finite-dimensional part of the paper is coherent and useful: the explicit formula in Theorem 1, the special-case recoveries, the comparison theorem via the Araki-Lieb-Thirring inequality, and the Riemannian submersion interpretation are all substantive and appear to be correct. The paper also supplies full proofs for the main finite-dimensional statements, and the RKHS formulas, if fully justified, would be practically valuable for kernel methods. The main weakness is in the infinite-dimensional part: the claimed unification with the Log-Hilbert-Schmidt distance on the full space PC2(H) is not actually established, because the distance is defined only on fixed scalar fibers. For these reasons the central claim needs substantial revision or extension.","major_comments":[{"comment":"The claimed infinite-dimensional unification with the Log-Hilbert-Schmidt distance is only proved within a fixed fiber, not on the full set PC2(H). Definition 3, Theorem 11, and Theorem 13 define d^α_proHS only for pairs (A+γI) and (B+γI) with the same γ, i.e. inside PC2(H)(γ) = {A+γI : A ∈ HS(H)}. For γ≠ν, the difference (γ−ν)I is not Hilbert-Schmidt, and Remark 2 explicitly defers this case to future work. The Log-Hilbert-Schmidt distance in Eq. (43), however, is defined on all of PC2(H), including pairs with different scalar parts. Consequently Theorem 12 recovers the Log-Hilbert-Schmidt distance only for operators sharing the same γ, and the abstract's statement that the family generalizes to \"the set of positive definite Hilbert-Schmidt operators\" and recovers the Log-Hilbert-Schmidt distance as a special case is overbroad. The manuscript should either extend the construction to γ≠ν or explicitly reframe the infinite-dimensional contribution as a fiberwise unification; the same caveat affects the γ-dependent Gaussian-measure distances in Theorems 15 and 16 and Eq. (71).","section":"§3, Definition 3, Theorem 11, Theorem 13, Remark 2"},{"comment":"The passage from the repeated-row 3×3 block matrix in Eq. (74) to the limiting block matrix used in the proof of Corollary 5 is not justified. In Eq. (74) the block matrix has identical second and third rows, and this structure is repeated in Eqs. (79) and (97). In the proof of Corollary 5, however, the displayed limit matrix is [[0,0,(A^*A)^{2α}A^*B(B^*B)^{2α−1}],[0,0,0],[0,0,B^*A(A^*A)^{2α−1}A^*B(B^*B)^{2α−1}]], which has a different zero pattern. The proof does not explain how the repeated-row matrix is transformed into this matrix, and Lemma 11 is stated only for the latter block form. Since the closed-form Gram-matrix formulas in Theorem 17 and Corollary 5 depend on this trace identity, the missing spectral argument is load-bearing for the RKHS section. Please either supply the missing equivalence or correct the displayed matrices.","section":"§3.4, Proposition 5, Corollary 5, Eqs. (74), (79), (97)"}],"minor_comments":[{"comment":"The displayed limit in Eq. (12) has an unmatched bracket in the trace expression; the bracket should be closed before the equality sign.","section":"Theorem 2, Eq. (12)"},{"comment":"In the proof of Lemma 4, the final displayed estimate says that the bound tends to zero \"as α → ∞\"; the intended limit is α → 0.","section":"Lemma 4, proof"},{"comment":"In the long displayed computation in the proof of Proposition 3, the term \"(I+V)^α(I+V)\" appears where \"(I+B)^α(I+V)\" is evidently intended.","section":"Proposition 3, proof"},{"comment":"Lemma 11 is stated for finite n×n matrices, but in the proof of Corollary 5 it is applied to block operators on the separable Hilbert space H1; the infinite-dimensional version needed for the proof should be stated and proved.","section":"Lemma 11, Corollary 5"},{"comment":"The proof of Theorem 3 is only one sentence for α≠0 and does not explicitly handle the limiting case α=0; the metric property of the Log-Euclidean distance should be cited there.","section":"Theorem 3, proof"}],"recommendation":"major_revision","confidential_remarks":"The finite-dimensional contribution is likely sound and publishable, but the infinite-dimensional part, which is a central advertised contribution, has a genuine scope gap: the Log-Hilbert-Schmidt recovery is only fiberwise. The RKHS derivation also needs a rigorous justification of the block-matrix limit. I would encourage the editor to ask the author to either extend the construction to cross-fiber pairs or revise the claims so that they match what is actually proved."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The finite-dimensional part of this paper is the real contribution. The Alpha Procrustes family is a clean one-parameter interpolation between Bures-Wasserstein (α=1/2, up to factor 2) and Log-Euclidean (α→0), and the paper proves it is a metric for every α≠0, gives the associated Riemannian submersion metric and explicit geodesics, and shows the Alpha Procrustes distance is strictly smaller than the power Euclidean distance for non-commuting arguments via the Araki-Lieb-Thirring inequality. The Gaussian-measure corollary and the RKHS Gram-matrix formulas extend the reach. That part is convincing and worth building on.\n\nThe soft spot is the infinite-dimensional generalization. The stress-test note is correct: the Alpha Procrustes distance is only defined for operators with the same scalar part γ. Definition 3, Theorem 11 and Theorem 13 all fix γ ahead of time. The fibers PC2(H)(γ) are disjoint, so no distance is defined across different γ's, while the Log-Hilbert-Schmidt distance in equation (43) is defined on all of PC2(H). The claimed recovery of Log-Hilbert-Schmidt as α→0 therefore only holds inside a fixed fiber, not on the claimed domain. Remark 2 openly defers γ≠ν to future work, so this is a genuine scope gap in the central infinite-dimensional contribution, not a missing line in a proof.\n\nThere are also presentation problems in the RKHS section: the 3x3 block matrices in equations (74), (79) and (97) show the second and third rows identical, which cannot be right since C23 differs from C22; the proof of Corollary 5 passes to a different block matrix; and there are minor typos such as the unbalanced bracket in Theorem 2 and a slip in Lemma 4's limit (it says α→∞ where the context is α→0). These are fixable, but they affect the advertised closed-form Gram-matrix formulas and need care.\n\nThe finite-dimensional story holds up. The infinite-dimensional story is plausible for fixed fibers, but the paper oversells it as a unification on all PC2(H). That should be either proved or explicitly narrowed.\n\nWho is this for? People working on SPD geometry, optimal transport between Gaussians, and kernel methods with covariance operators will get value from the finite-dimensional results and the fixed-fiber operator formulas. It deserves a serious referee; with the scope issue addressed and the RKHS matrices corrected, it would be a solid publication.\n\nRecommendation: send to peer review, require major revision. Point the authors at the cross-γ gap first.","headline":"A genuine finite-dimensional unification; the infinite-dimensional claim needs a scope correction before the paper can stand.","tokens_in":30818,"tokens_out":2425,"would_cite":true,"duration_ms":23406,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["15B48","46E22","47B10"],"pacs":[],"model":"deepseek-v4-flash","headline":"A parametrized family of distances, the Alpha Procrustes distances, unifies the Bures-Wasserstein and Log-Euclidean distances on positive definite operators, and extends to infinite-dimensional Hilbert-Schmidt operators.","keywords":["Alpha Procrustes distance","Bures-Wasserstein distance","Log-Euclidean distance","Log-Hilbert-Schmidt distance","positive definite operators","Gaussian measures","reproducing kernel Hilbert space","Riemannian submersion"],"falsifier":"Produce an invertible operator $I+A$ with $A$ Hilbert-Schmidt whose polar unitary factor $I+U$ has $U$ not Hilbert-Schmidt; such an example would falsify Lemma 1 and thereby invalidate the trace formula (52) for the infinite-dimensional Alpha Procrustes distance.","tokens_in":29776,"feed_emoji":"📐","tokens_out":13892,"duration_ms":109361,"temperature":0.7,"pith_summary":"The paper introduces a one-parameter family of distances between positive definite matrices, called Alpha Procrustes distances, defined by aligning matrix powers up to unitary rotations. It proves that at parameter α=1/2 the distance is exactly twice the Bures-Wasserstein (optimal-transport) distance, and that as α→0 it converges to the Log-Euclidean distance. The same construction is carried over to positive definite unitized Hilbert-Schmidt operators on a Hilbert space, where it recovers the Bures-Wasserstein and Log-Hilbert-Schmidt distances as special cases, and to Gaussian measures, where it generalizes the L2-Wasserstein distance. In reproducing kernel Hilbert spaces the distances all have closed forms in terms of kernel Gram matrices. If correct, the paper shows that two previously separate geometries of covariance matrices are the endpoints of one continuous family with explicit formulas.","feed_headline":"One distance family unifies Wasserstein and Log-Euclidean metrics","feed_subtitle":"A parameter α slides between optimal-transport and Log-Euclidean geometry, also for Hilbert-Schmidt operators.","key_machinery":"The central object is the Alpha Procrustes distance, defined by the unitary alignment problem $d^{\\alpha}_{\\mathrm{proE}}(A,B)=\\min_{U}\\|(A^{\\alpha}-B^{\\alpha}U)/\\alpha\\|_{F}$ and given in closed form by the trace expression in Eq. (8). Three mechanisms carry the argument. First, the polar decomposition of $B^{\\alpha}A^{\\alpha}$ solves the Procrustes problem and produces the explicit trace formula. Second, the maps $\\pi_{\\alpha}(G)=(\\alpha^{2}GG^{*})^{1/(2\\alpha)}$ from the general linear group to the SPD cone become Riemannian submersions under the metric (29), so geodesics and distances descend from straight lines in $\\mathrm{GL}(n)$, which is how the geodesic formula (34) and the metric property arise. Third, in infinite dimensions the algebra of unitized Hilbert-Schmidt operators $\\mathrm{HS}_{\\mathrm{X}}(H)=\\mathrm{HS}(H)+\\mathbb{R}I$ and Lemma 1 (polar decomposition stays inside the unitized class) keep the trace expressions finite. The Araki-Lieb-Thirring inequality supplies the comparison with power-Euclidean distances.","core_discovery":"At the core is the closed-form solution of the Alpha Procrustes optimization problem: $d^{\\alpha}_{\\mathrm{proE}}(A,B)=\\frac{1}{|\\alpha|}\\min_{U\\in\\mathrm{U}(n)}\\|A^{\\alpha}-B^{\\alpha}U\\|_{F}=\\big(\\frac{1}{\\alpha^{2}}\\mathrm{tr}[A^{2\\alpha}+B^{2\\alpha}-2(A^{\\alpha}B^{2\\alpha}A^{\\alpha})^{1/2}]\\big)^{1/2}$ for every nonzero α. The paper proves that this is a metric on the SPD cone, equal to twice the Bures-Wasserstein distance at α=1/2 and to $\\|\\log A-\\log B\\|_{F}$ as α→0. It then shows that these distances are the Riemannian distances of explicit Riemannian metrics on the SPD manifold, obtained as Riemannian submersions from the general linear group with the Frobenius metric, with the Wasserstein Riemannian metric and the Log-Euclidean metric as special cases. In infinite dimensions, the family is defined on unitized positive definite Hilbert-Schmidt operators, where the α→0 limit is the Log-Hilbert-Schmidt distance and trace-class operators with α≥1/2 recover the Bures-Wasserstein distance; in the RKHS setting all distances reduce to formulas in kernel Gram matrices.","pith_inferences":["If the family is geodesically complete for every α, it would give a principled way to average covariance matrices while sweeping a trade-off between Wasserstein and Log-Euclidean behavior, which is currently a modeling choice in diffusion-tensor imaging and covariance tracking.","The Gram-matrix formulations suggest all RKHS distances in the family can be evaluated from $m\\times m$ kernels in $O(m^3)$ time, making the parameter α cheap to select by cross-validation or likelihood.","The submersion construction is not tied to the symmetric-positive cone; the same pattern may define analogous Alpha Procrustes metrics on other homogeneous spaces with polar decompositions, such as Grassmannians or flag manifolds.","The paper leaves open the case $\\gamma\\neq\\nu$ for two different unitization scales in infinite dimensions; a closed form there would complete the unification and is the natural next step."],"forward_implications":["There is a one-parameter family of Riemannian metrics on the SPD cone whose geodesic distances interpolate between the Bures-Wasserstein and Log-Euclidean geometries, so computations set up in one geometry can be continuously deformed to the other.","The same family defines metrics on positive definite unitized Hilbert-Schmidt operators, with the Log-Hilbert-Schmidt distance as the α→0 limit and the Bures-Wasserstein distance recovered for trace-class operators when $\\alpha\\geq 1/2$.","For Gaussian measures on Euclidean and separable Hilbert spaces, the family yields a parametrized set of metrics containing the squared L2-Wasserstein distance at α=1/2 and a Log-Euclidean-type distance as α→0.","In RKHS settings, all members of the family between empirical covariance operators are computable in closed form from the kernel Gram matrices K[X], K[Y], and K[X,Y], without ever constructing the covariance operators explicitly.","For any fixed exponent α, the Alpha Procrustes distance is never larger than the power-Euclidean distance with the same α, and the two coincide exactly when the matrices commute."],"supporting_citations":[{"why":"Supplies the Procrustes formulation of the Bures-Wasserstein distance and the horizontal-lifting argument used to prove the geodesic formula.","marker":"[3]"},{"why":"Gives the closed-form Fréchet/Bures-Wasserstein distance between multivariate normals that is recovered at α=1/2.","marker":"[4]"},{"why":"Extends the L2-Wasserstein distance formula to Hilbert spaces, the infinite-dimensional setting addressed in Theorem 9.","marker":"[6]"},{"why":"Introduces Procrustes metrics on covariance operators and their link to optimal transport of Gaussian processes.","marker":"[11]"},{"why":"Defines the Log-Euclidean metric on SPD matrices, the α→0 endpoint of the new family.","marker":"[2]"},{"why":"Introduces the Log-Hilbert-Schmidt distance on positive definite unitized operators, recovered as α→0 in Theorem 12.","marker":"[12]"},{"why":"Establishes the unitized Hilbert-Schmidt space HSX(H) and its geometry, the setting for the infinite-dimensional distances.","marker":"[8]"},{"why":"The Araki-Lieb-Thirring inequality used in Theorem 4 to compare Alpha Procrustes with power-Euclidean distances.","marker":"[9]"},{"why":"Provides the operator identities expressing powers of (AA^*+I) through A^*A, used in the RKHS Gram-matrix formulas.","marker":"[14]"},{"why":"Gives the Wasserstein Riemannian metric on Gaussian densities, recovered as the α=1/2 case of the submersion metric.","marker":"[19]"}],"fun_headline_variants":["A parameter α slides between Wasserstein and Log-Euclidean metrics","New distances unify optimal transport and Log-Euclidean geometry","Alpha Procrustes: one family, two classic SPD metrics","From Bures-Wasserstein to Log-Euclidean via a single formula","α-Procrustes: a bridge between Wasserstein and Log-Euclidean geometry"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The infinite-dimensional construction rests on Lemma 1, which asserts that the polar decomposition of any invertible unitized Hilbert-Schmidt operator $A+\\gamma I$ has unitary factor $I+R$ with $R$ Hilbert-Schmidt; if the unitary factor could leave the unitized Hilbert-Schmidt class, the explicit Alpha Procrustes distance on $\\mathrm{PC}_2(H)$ would not be finite.","fun_headline_variants_meta":{"raw":{"variants":["A parameter α slides between Wasserstein and Log-Euclidean metrics","New distances unify optimal transport and Log-Euclidean geometry","Alpha Procrustes: one family, two classic SPD metrics","From Bures-Wasserstein to Log-Euclidean via a single formula","α-Procrustes: a bridge between Wasserstein and Log-Euclidean geometry"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0011,"raw_usage":{"total_tokens":4656,"prompt_tokens":1078,"completion_tokens":3578,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":694,"completion_tokens_details":{"reasoning_tokens":3482}},"tokens_in":694,"tokens_out":3578,"duration_ms":25583,"temperature":1.0,"reasoning_tokens":3482,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:17:07.257763+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Produce an invertible operator $I+A$ with $A$ Hilbert-Schmidt whose polar unitary factor $I+U$ has $U$ not Hilbert-Schmidt; such an example would falsify Lemma 1 and thereby invalidate the trace formula (52) for the infinite-dimensional Alpha Procrustes distance.","supporting_citations":[{"cited_title":"Expositiones Mathematicae (2018)","cited_arxiv_id":null,"evidence_quote":"Supplies the Procrustes formulation of the Bures-Wasserstein distance and the horizontal-lifting argument used to prove the geodesic formula."},{"cited_title":"Journal of Multivariate Analysis 12(3), 450 – 455 (1982)","cited_arxiv_id":null,"evidence_quote":"Gives the closed-form Fréchet/Bures-Wasserstein distance between multivariate normals that is recovered at α=1/2."},{"cited_title":"Mathematische Nachrichten 147(1), 185–203 (1990)","cited_arxiv_id":null,"evidence_quote":"Extends the L2-Wasserstein distance formula to Hilbert spaces, the infinite-dimensional setting addressed in Theorem 9."},{"cited_title":"San khya A pp","cited_arxiv_id":null,"evidence_quote":"Introduces Procrustes metrics on covariance operators and their link to optimal transport of Gaussian processes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the Log-Euclidean metric on SPD matrices, the α→0 endpoint of the new family."},{"cited_title":"In: Advances in Ne ural Information Pro- cessing Systems 27 (NIPS 2014), pp","cited_arxiv_id":null,"evidence_quote":"Introduces the Log-Hilbert-Schmidt distance on positive definite unitized operators, recovered as α→0 in Theorem 12."},{"cited_title":"Differential Geometry and its Applications 25, 679–700 (2007)","cited_arxiv_id":null,"evidence_quote":"Establishes the unitized Hilbert-Schmidt space HSX(H) and its geometry, the setting for the infinite-dimensional distances."},{"cited_title":"In: Lieb, E., S.B., Wrightman, A","cited_arxiv_id":null,"evidence_quote":"The Araki-Lieb-Thirring inequality used in Theorem 4 to compare Alpha Procrustes with power-Euclidean distances."},{"cited_title":"In: Information Geo metry and its Appli- cations IV","cited_arxiv_id":null,"evidence_quote":"Provides the operator identities expressing powers of (AA^*+I) through A^*A, used in the RKHS Gram-matrix formulas."},{"cited_title":"Osaka Journal of Math- ematics 48(4), 1005–1026 (2011)","cited_arxiv_id":null,"evidence_quote":"Gives the Wasserstein Riemannian metric on Gaussian densities, recovered as the α=1/2 case of the submersion metric."}],"review_version":1}