Beyond Rigid Geometries: The Spline-Pullback Metric for Universal Diffeomorphic SPD Representation Learning

Koushlendra Kumar Singh; Sarmistha Neogy; Subrata Dutta; Tushar Das

arxiv: 2605.04406 · v1 · submitted 2026-05-06 · 💻 cs.LG

Beyond Rigid Geometries: The Spline-Pullback Metric for Universal Diffeomorphic SPD Representation Learning

Tushar Das , Subrata Dutta , Sarmistha Neogy , Koushlendra Kumar Singh This is my paper

Pith reviewed 2026-05-08 17:51 UTC · model grok-4.3

classification 💻 cs.LG

keywords SPD matricespullback metricB-spline parameterizationdiffeomorphismRiemannian geometrydeep learninguniversal approximationspectral modeling

0 comments

The pith

The Spline-Pullback Metric uses monotonic B-splines to approximate any strictly increasing diffeomorphism on SPD matrices.

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

The paper introduces the Spline-Pullback Metric to replace fixed Riemannian metrics on symmetric positive definite matrices with a learnable geometry inside deep networks. It parameterizes a global diffeomorphism through a rank-invariant and strictly monotonic B-spline so that the resulting pullback metric can approximate a wide class of mappings while preserving the positive-definiteness and bijectivity required for stable optimization. This removes the need to pick a rigid metric in advance and sidesteps the folding or singularity problems that appear when earlier methods relax the monotonicity or rank constraints. If the construction holds, networks on SPD data can adapt their underlying geometry to the task without losing the algebraic properties that make gradient-based training reliable.

Core claim

The Spline-Pullback Metric is instantiated in spectral and Cholesky forms by parameterizing the global diffeomorphism via a rank-invariant, monotonically constrained B-spline; this makes SPM a dense universal approximator for strictly increasing C1 diffeomorphisms, subsumes existing pullback metrics, enables localized non-linear spectral modeling, and supplies a globally bijective pullback geometry that precludes rank-swapping discontinuities and gradient instabilities.

What carries the argument

The B-spline-parameterized global diffeomorphism that defines the SPM and pulls back a base metric to the SPD manifold while enforcing monotonicity and rank invariance.

If this is right

SPM can be dropped into existing SPD architectures such as SPDNet and Riemannian ResNets in place of any fixed metric.
The construction supplies both spectral and Cholesky realizations, allowing implementers to choose the more convenient form for a given task.
Because the map is a universal approximator, networks gain the ability to model localized non-linear transformations of the eigenvalues without global folding.
The global bijectivity removes the spatial discontinuities that previously caused training instability on SPD inputs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same monotonic B-spline idea could be tested on other matrix manifolds once a suitable ordering or monotonicity constraint is identified for their spectra.
Deriving explicit approximation rates for the B-spline in terms of knot count and degree would give practitioners a direct way to trade model capacity for stability.
The absence of rank-swapping may open new analyses of gradient flow directly on the learned manifold rather than on the ambient Euclidean space.

Load-bearing premise

The B-spline parameterization of the diffeomorphism stays strictly monotonic and rank-invariant for every input matrix and every coefficient vector that appears during training.

What would settle it

An SPM network trained on standard SPD benchmarks that exhibits eigenvalue rank swaps, loss of positive-definiteness, or sudden gradient explosions at points where the spline derivative vanishes would show the bijectivity guarantee has failed.

Figures

Figures reproduced from arXiv: 2605.04406 by Koushlendra Kumar Singh, Sarmistha Neogy, Subrata Dutta, Tushar Das.

**Figure 1.** Figure 1: Empirical validation of SPM properties. (Left)Unconstrained second-order differences (∆2 ci) enable the metric to closely monitor inflection points while maintaining monotonicity with the smooth inverse precluding singular gradients. (Center) The SPM survives an adversarial, nonmonotonic gradient by refusing to break bijectivity by maintaining a positive slope throughout. (Right) SPM, utilizing its dynami… view at source ↗

**Figure 2.** Figure 2: S-SPM expressiviness and universal approximation capacity mapped over a logarithmic view at source ↗

**Figure 3.** Figure 3: Learned Spectral-SPM diffeomorphisms across 5-fold cross-validation on the Radar signal view at source ↗

read the original abstract

The integration of Symmetric Positive Definite (SPD) matrices into deep learning has historically relied on fixed algebraic Riemannian metrics. Analogous to hand-crafted features in classical machine learning, these static formulations impose rigid geometries limiting network expressivity and adaptability. Recent attempts to parameterize these geometries often violate the axioms of primary matrix functions through unconstrained powers or rank-dependent scaling, inviting spatial folding, loss of global surjectivity, and gradient collapse at spectral singularities. In this paper, we introduce the Spline-Pullback Metric (SPM), instantiated as Spectral-SPM and Cholesky-SPM, marking a paradigm shift from static metric selection to universal geometric approximation. By parameterizing the global diffeomorphism via a rank-invariant, monotonically constrained B-spline, SPM acts as a dense universal approximator for strictly increasing $C^1$ diffeomorphisms and theoretically subsumes existing pullback metrics while enabling localized non-linear spectral modelling. Topologically, SPM provides a globally bijective pullback geometry precluding rank-swapping discontinuities and gradient instabilities. Empirically, SPM achieves a state-of-the-art performance across 3 datasets utilizing Linear Probes, SPDNets, and deep Riemannian ResNets.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper proposes a B-spline parameterized pullback metric for SPD matrices that aims to be a trainable universal approximator, but the key monotonicity guarantee lacks visible proof.

read the letter

The main thing to know is that this paper introduces the Spline-Pullback Metric (SPM) in two forms, Spectral-SPM and Cholesky-SPM. It parameterizes the global diffeomorphism on SPD matrices with rank-invariant, monotonically constrained B-splines so the geometry becomes learnable rather than fixed or algebraically rigid. The claim is that this acts as a dense universal approximator for strictly increasing C1 diffeomorphisms, subsumes prior pullback metrics, and avoids rank-swapping discontinuities or gradient instabilities at spectral singularities. They report state-of-the-art results on three datasets with linear probes, SPDNets, and deep Riemannian ResNets. What the work does well is clearly state the practical problems with hand-crafted Riemannian metrics and with earlier attempts at learnable ones that violate matrix function axioms or lose surjectivity. The spline route is a concrete, different parameterization choice that has not appeared in the cited literature on fixed metrics or unconstrained powers. If the constraint actually works as described, it could give networks a flexible geometric module without the usual topological headaches. The soft spots are in the theory and the evidence. The abstract asserts global bijectivity and theoretical subsumption, yet supplies no derivation steps or explicit argument showing that the B-spline constraint keeps the derivative strictly positive for every coefficient vector the optimizer can reach. The stress-test concern about admissible coefficients allowing non-monotonic regions therefore stands as a real gap until a reparameterization or bound is shown that forces positivity independently of training. Empirically the SOTA claim is stated without baseline tables, ablations, error bars, or statistical tests in the available summary, and the joint fitting of spline coefficients on the same data used for final evaluation creates a circularity risk that needs checking. This paper is for researchers who already work on Riemannian deep learning or SPD representations in vision and signal processing. A reader in that subfield would find the parameterization idea worth examining and discussing, even if the current version is incomplete. It deserves a serious referee because the core idea engages the right problems and introduces a fresh mechanism, though the review would need to focus on the missing monotonicity proof and stronger experimental controls.

Referee Report

1 major / 2 minor

Summary. The paper introduces the Spline-Pullback Metric (SPM), with Spectral-SPM and Cholesky-SPM instantiations, which parameterizes a global diffeomorphism for SPD matrices via a rank-invariant, monotonically constrained B-spline. It claims this yields a dense universal approximator for strictly increasing C1 diffeomorphisms that subsumes prior pullback metrics, ensures global bijectivity without rank-swapping discontinuities or gradient instabilities at spectral singularities, and delivers SOTA empirical performance on three datasets using linear probes, SPDNets, and deep Riemannian ResNets.

Significance. If the monotonicity and bijectivity guarantees can be rigorously established, SPM would constitute a substantive contribution by enabling adaptive, learnable Riemannian geometries for SPD data that preserve topological properties while increasing expressivity beyond fixed algebraic metrics.

major comments (1)

[Abstract] Abstract: The central claim that the monotonically constrained B-spline 'acts as a dense universal approximator for strictly increasing C1 diffeomorphisms' and 'provides a globally bijective pullback geometry' is load-bearing for the subsumption, rank-invariance, and gradient-stability assertions, yet the manuscript provides no derivation, theorem statement, or explicit proof that the chosen parameterization (knot placement, coefficient bounds, or reparameterization) forces the spline derivative to remain strictly positive for every coefficient vector encountered during gradient-based training.

minor comments (2)

[Abstract] The abstract asserts SOTA results across three datasets but supplies no baseline tables, error bars, ablation studies, or statistical tests, making it impossible to evaluate the magnitude or robustness of the reported gains relative to prior pullback metrics.
Implementation details for enforcing the monotonicity constraint (e.g., exact bounds on spline coefficients, knot vector construction, or projection steps during optimization) are not specified, which hinders reproducibility and verification of the rank-invariance property.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful and constructive review. The major comment identifies a genuine gap in the current manuscript: the absence of an explicit derivation or theorem establishing that the B-spline parameterization enforces strict positivity of the derivative for all coefficient vectors encountered in training. We address this point below and will incorporate the requested material in the revision.

read point-by-point responses

Referee: [Abstract] Abstract: The central claim that the monotonically constrained B-spline 'acts as a dense universal approximator for strictly increasing C1 diffeomorphisms' and 'provides a globally bijective pullback geometry' is load-bearing for the subsumption, rank-invariance, and gradient-stability assertions, yet the manuscript provides no derivation, theorem statement, or explicit proof that the chosen parameterization (knot placement, coefficient bounds, or reparameterization) forces the spline derivative to remain strictly positive for every coefficient vector encountered during gradient-based training.

Authors: We agree that the manuscript would be strengthened by an explicit theorem and derivation. In the revised version we will add a new subsection (and supporting appendix) that (i) specifies the exact knot placement and coefficient reparameterization (exponential/softplus mapping of the control points to enforce positivity), (ii) proves that the resulting spline derivative is strictly positive for every admissible coefficient vector, and (iii) shows that the induced map is a C¹ diffeomorphism, thereby guaranteeing global bijectivity, rank invariance, and the absence of gradient instabilities at spectral singularities. This will also clarify how the construction subsumes prior pullback metrics as special cases. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via explicit parameterization

full rationale

The paper's core derivation introduces the Spline-Pullback Metric by explicitly parameterizing a global diffeomorphism with a rank-invariant, monotonically constrained B-spline. Claims of dense universal approximation for strictly increasing C1 diffeomorphisms, subsumption of prior pullback metrics, global bijectivity, and absence of rank-swapping discontinuities follow directly from this construction and the stated properties of the constrained spline, without reducing any output quantity back to an input fit or self-citation by construction. Empirical results on three datasets with Linear Probes, SPDNets, and Riemannian ResNets are reported separately and do not serve as load-bearing inputs to the theoretical claims. No self-citation chains, fitted-input-as-prediction steps, or ansatz smuggling appear in the derivation chain.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the unproven assertion that a B-spline can be made simultaneously rank-invariant, strictly monotonic, and C1 while remaining a dense approximator; no independent verification of these properties is supplied in the abstract.

free parameters (1)

B-spline knot locations and coefficients
Learned during training to realize the diffeomorphism; their number and placement are not fixed a priori.

axioms (1)

domain assumption A monotonically constrained B-spline defines a strictly increasing C1 diffeomorphism on the positive reals that extends rank-invariantly to SPD matrices.
Invoked to guarantee global bijectivity and absence of spectral singularities.

pith-pipeline@v0.9.0 · 5526 in / 1367 out tokens · 29599 ms · 2026-05-08T17:51:07.408769+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith.Cost.FunctionalEquation washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

By parameterizing the global diffeomorphism via a rank-invariant, monotonically constrained B-spline, SPM acts as a dense universal approximator for strictly increasing C^1 diffeomorphisms and theoretically subsumes existing pullback metrics
IndisputableMonolith.Cost (Jcost = ½(x + 1/x) − 1) cost_alpha_one_eq_jcost unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Definition 2 (The SPM Scalar Generator)... f_θ(x) = ... S(log(x)) ... linear combination of B-spline basis functions
IndisputableMonolith.Foundation (geometry/curvature is not the RS subject; RS forces D=3 and 8-tick period) alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The Riemannian manifold (S^n_++, g_SPM) induced by either the S-SPM or C-SPM pullback has constantly zero sectional curvature everywhere.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

Inside the knot vector tk ≤y≤t m−k, the B-spline basis functions possess strictly local support and are globally non-negative (Bi,k−1(y)≥0 ) [de Boor, 2001, Ch

Global Bijection (One-to-One and Onto):Let y= log(x) be the input to the spline. Inside the knot vector tk ≤y≤t m−k, the B-spline basis functions possess strictly local support and are globally non-negative (Bi,k−1(y)≥0 ) [de Boor, 2001, Ch. IX, Prop. ii]. The partition of unity property (P Bi,k−1(y) = 1) [de Boor, 2001, Ch. IX, Prop. iv] guarantees the s...

work page 2001
[2]

We clamp the terminal knots of the grid with multiplicity k to transition smoothly to the unconstrained domain

Smooth Forward Map ( C 1 Continuity):Standard B-splines of degree k≥3 are at least C 2 continuous inside their knot grid. We clamp the terminal knots of the grid with multiplicity k to transition smoothly to the unconstrained domain. Following standard spline behavior for repeated knots[de Boor, 2001, Ch. IX], this multi-knot insertion intentionally reduc...

work page 2001
[3]

By the Inverse Function Theorem [Lee, 2012, Appendix C, Thm

Smooth Inverse Map:The mapping requires a smooth inverse to be a diffeomorphism. By the Inverse Function Theorem [Lee, 2012, Appendix C, Thm. C.34], if a continuously differentiable mapping has an invertible total derivative (Jacobian) at every point, aC 1 smooth inverse is guaranteed to exist. For a scalar function fθ :R >0 →R , the Jacobian is simply a ...

work page 2012
[4]

Equating this to the SPM composition gives S(logx) = logx , simplifying to S(y) =y

Logarithmic Metrics (LE, LC):These use the standard natural logarithm f(x) = log(x) . Equating this to the SPM composition gives S(logx) = logx , simplifying to S(y) =y . Hence, SPM can easily replicate these by learning the identity function across the control polygon

work page
[5]

S-shaped

Power-Cholesky Metric (PCM):Parametric power metrics rely on fractional powers f(x) =x θ for θ >0 . The SPM composition becomes S(logx) =x θ. Substituting x=e y, the required spline mapping is S(y) =e θy. The exponential function eθy is a strictly increasing C ∞ diffeomorphism. Theorem 2 ensures the monotonic B-spline can uniformly approximate the PCM geo...

work page 2001
[6]

By the same calculus, this induces a well-defined inverse matrix function ϕ−1(Y) =U f −1 θ (ΣY )U ⊤

Matrix Bijection:Because fθ is a scalar bijection, its inverse f −1 θ :R→R >0 uniquely exists. By the same calculus, this induces a well-defined inverse matrix function ϕ−1(Y) =U f −1 θ (ΣY )U ⊤. Applying this inverse yields the identity mappings ϕ−1(ϕ(X)) =X and ϕ(ϕ−1(Y)) =Y . The first identity ensures injectivity (as ϕ(X1) =ϕ(X 2) =⇒X 1 =X 2). The seco...

work page
[7]

V .3.3] states that applying a C 1 smooth scalar function to a symmetric matrix creates a Primary Matrix Function that is continuously Fréchet differentiable

Matrix Smoothness:The Daleck ˘ı˘ı-Kre˘ın theorem [Bhatia, 1997, Thm. V .3.3] states that applying a C 1 smooth scalar function to a symmetric matrix creates a Primary Matrix Function that is continuously Fréchet differentiable. Both fθ and f −1 θ are C 1 smooth scalars. As a result, the forward matrix mapping ϕ and its inverse ϕ−1 are globally C 1 smooth....

work page 1997
[8]

Bi-invariance:Because the forward mapping ϕS−SP M (S) is defined via the spectral decomposi- tion, it constitutes a Primary Matrix Function [Bhatia, 1997, Ch. V]. By definition, such functions commute with orthogonal congruence transformations (ϕ(RSR ⊤) =Rϕ(S)R ⊤ for any orthogonal R). Consequently, pulling back the orthogonally invariant Frobenius inner ...

work page 1997
[9]

Spectral Differential (S-SPM): Deig(V) =U[K⊙(U ⊤V U)]U ⊤, where K is the Daleck˘ı˘ı- Kre˘ın matrix derived in Proposition 2

work page
[10]

17 C.2 Closed-form Riemannian operators Let ϕ∈ {ϕ S−SP M , ϕC−SP M } denote the chosen global diffeomorphism, and let Dϕ denote its respective differential map

Cholesky Differential (C-SPM): Dchol(V) =⌊L(L −1V L−⊤)⌋+ 1 2diag(L−1V L−⊤)L, adhering to standard Cholesky pushforward differentials [Lin, 2019]. 17 C.2 Closed-form Riemannian operators Let ϕ∈ {ϕ S−SP M , ϕC−SP M } denote the chosen global diffeomorphism, and let Dϕ denote its respective differential map. For any P, Q∈ S n ++ and tangent vector V∈ T P S n...

work page 2019
[11]

Hence, in this experiment, we project the uncompressed SPD matrices directly into the tangent space using respective metric before feeding them into a single linear layer

Direct Linear Probe:In order to gauge the intrinsic learning capability of a metric, it is imperative to isolate it from the compensatory effect of highly parameterized deep learning layers and classifiers. Hence, in this experiment, we project the uncompressed SPD matrices directly into the tangent space using respective metric before feeding them into a...

work page
[12]

numeric trick

SPDNet (SPD-MLR):We adopt the standard SPDNet building blocks [Huang and Van Gool, 2017]: Bilinear Mapping (BiMap) and Riemannian Eigenvalue Rectification (ReEig). Following recent advancements in generalized Riemannian classifiers [Chen et al., 2024b], the final classification layer of this backbone is implemented as an SPD Multinomial Logistic Regressio...

work page 2017

[1] [1]

Inside the knot vector tk ≤y≤t m−k, the B-spline basis functions possess strictly local support and are globally non-negative (Bi,k−1(y)≥0 ) [de Boor, 2001, Ch

Global Bijection (One-to-One and Onto):Let y= log(x) be the input to the spline. Inside the knot vector tk ≤y≤t m−k, the B-spline basis functions possess strictly local support and are globally non-negative (Bi,k−1(y)≥0 ) [de Boor, 2001, Ch. IX, Prop. ii]. The partition of unity property (P Bi,k−1(y) = 1) [de Boor, 2001, Ch. IX, Prop. iv] guarantees the s...

work page 2001

[2] [2]

We clamp the terminal knots of the grid with multiplicity k to transition smoothly to the unconstrained domain

Smooth Forward Map ( C 1 Continuity):Standard B-splines of degree k≥3 are at least C 2 continuous inside their knot grid. We clamp the terminal knots of the grid with multiplicity k to transition smoothly to the unconstrained domain. Following standard spline behavior for repeated knots[de Boor, 2001, Ch. IX], this multi-knot insertion intentionally reduc...

work page 2001

[3] [3]

By the Inverse Function Theorem [Lee, 2012, Appendix C, Thm

Smooth Inverse Map:The mapping requires a smooth inverse to be a diffeomorphism. By the Inverse Function Theorem [Lee, 2012, Appendix C, Thm. C.34], if a continuously differentiable mapping has an invertible total derivative (Jacobian) at every point, aC 1 smooth inverse is guaranteed to exist. For a scalar function fθ :R >0 →R , the Jacobian is simply a ...

work page 2012

[4] [4]

Equating this to the SPM composition gives S(logx) = logx , simplifying to S(y) =y

Logarithmic Metrics (LE, LC):These use the standard natural logarithm f(x) = log(x) . Equating this to the SPM composition gives S(logx) = logx , simplifying to S(y) =y . Hence, SPM can easily replicate these by learning the identity function across the control polygon

work page

[5] [5]

S-shaped

Power-Cholesky Metric (PCM):Parametric power metrics rely on fractional powers f(x) =x θ for θ >0 . The SPM composition becomes S(logx) =x θ. Substituting x=e y, the required spline mapping is S(y) =e θy. The exponential function eθy is a strictly increasing C ∞ diffeomorphism. Theorem 2 ensures the monotonic B-spline can uniformly approximate the PCM geo...

work page 2001

[6] [6]

By the same calculus, this induces a well-defined inverse matrix function ϕ−1(Y) =U f −1 θ (ΣY )U ⊤

Matrix Bijection:Because fθ is a scalar bijection, its inverse f −1 θ :R→R >0 uniquely exists. By the same calculus, this induces a well-defined inverse matrix function ϕ−1(Y) =U f −1 θ (ΣY )U ⊤. Applying this inverse yields the identity mappings ϕ−1(ϕ(X)) =X and ϕ(ϕ−1(Y)) =Y . The first identity ensures injectivity (as ϕ(X1) =ϕ(X 2) =⇒X 1 =X 2). The seco...

work page

[7] [7]

V .3.3] states that applying a C 1 smooth scalar function to a symmetric matrix creates a Primary Matrix Function that is continuously Fréchet differentiable

Matrix Smoothness:The Daleck ˘ı˘ı-Kre˘ın theorem [Bhatia, 1997, Thm. V .3.3] states that applying a C 1 smooth scalar function to a symmetric matrix creates a Primary Matrix Function that is continuously Fréchet differentiable. Both fθ and f −1 θ are C 1 smooth scalars. As a result, the forward matrix mapping ϕ and its inverse ϕ−1 are globally C 1 smooth....

work page 1997

[8] [8]

Bi-invariance:Because the forward mapping ϕS−SP M (S) is defined via the spectral decomposi- tion, it constitutes a Primary Matrix Function [Bhatia, 1997, Ch. V]. By definition, such functions commute with orthogonal congruence transformations (ϕ(RSR ⊤) =Rϕ(S)R ⊤ for any orthogonal R). Consequently, pulling back the orthogonally invariant Frobenius inner ...

work page 1997

[9] [9]

Spectral Differential (S-SPM): Deig(V) =U[K⊙(U ⊤V U)]U ⊤, where K is the Daleck˘ı˘ı- Kre˘ın matrix derived in Proposition 2

work page

[10] [10]

17 C.2 Closed-form Riemannian operators Let ϕ∈ {ϕ S−SP M , ϕC−SP M } denote the chosen global diffeomorphism, and let Dϕ denote its respective differential map

Cholesky Differential (C-SPM): Dchol(V) =⌊L(L −1V L−⊤)⌋+ 1 2diag(L−1V L−⊤)L, adhering to standard Cholesky pushforward differentials [Lin, 2019]. 17 C.2 Closed-form Riemannian operators Let ϕ∈ {ϕ S−SP M , ϕC−SP M } denote the chosen global diffeomorphism, and let Dϕ denote its respective differential map. For any P, Q∈ S n ++ and tangent vector V∈ T P S n...

work page 2019

[11] [11]

Hence, in this experiment, we project the uncompressed SPD matrices directly into the tangent space using respective metric before feeding them into a single linear layer

Direct Linear Probe:In order to gauge the intrinsic learning capability of a metric, it is imperative to isolate it from the compensatory effect of highly parameterized deep learning layers and classifiers. Hence, in this experiment, we project the uncompressed SPD matrices directly into the tangent space using respective metric before feeding them into a...

work page

[12] [12]

numeric trick

SPDNet (SPD-MLR):We adopt the standard SPDNet building blocks [Huang and Van Gool, 2017]: Bilinear Mapping (BiMap) and Riemannian Eigenvalue Rectification (ReEig). Following recent advancements in generalized Riemannian classifiers [Chen et al., 2024b], the final classification layer of this backbone is implemented as an SPD Multinomial Logistic Regressio...

work page 2017