Geodesic Flow Matching for Denoising High-Dimensional Structured Representations¶

Conference: ICML 2026
arXiv: 2606.00248
Code: https://github.com/kremHabashy/CleanupSSP
Area: Representation Learning / Flow Matching / Neuro-symbolic / Manifold Geometry
Keywords: Geodesic Flow Matching, Spatial Semantic Pointer (SSP), Clifford Torus, Neuro-symbolic Cleanup, Spiking Neural SLAM

TL;DR¶

Focusing on high-dimensional structured representations like Spatial Semantic Pointers (SSPs) in Vector Symbolic Architectures—which are "embedded in a Clifford torus within a unit hypersphere"—the authors observe that Euclidean linear interpolation in standard Flow Matching passes through the sphere's interior, causing amplitude collapse and phase destruction. By using Log/Exp maps to constrain the flow to the sphere via Geodesic Flow Matching (GFM), they reduce path error in spiking neural SLAM by 72% and enable a 1500-neuron path integrator to match the accuracy of a 2500-neuron baseline.

Background & Motivation¶

Background: Vector Symbolic Architectures (VSA, focusing on Plate 95 HRR here) encode symbols into high-dimensional vectors, performing compositional reasoning via bundling (addition) and binding (circular convolution). Spatial Semantic Pointers (SSP) extend this by encoding continuous coordinates \(x\in\mathbb{R}^m\) into \(d>1000\) dimensional vectors using Fourier phases \(\tilde\phi(x)_j=e^{i\langle\theta_j, x\rangle}\). This creates continuous "position-vector" cognitive maps for path integration, SLAM, and hippocampal-entorhinal models. All VSA systems rely on a critical step—cleanup: projecting vectors contaminated by cross-talk, phase drift, or spiking noise back onto the valid representation manifold.

Limitations of Prior Work: Traditional cleanup follows two paths: (1) discrete prototypes (Hopfield networks), which are unsuitable for continuous SSPs; or (2) grid lookup + L-BFGS optimization, which either suffers from exponential grid resolution scaling or "snaps to the wrong prototype" under high noise. While recent works view diffusion/flow matching as modern continuous associative memories, directly applying them poses two issues: (a) diffusion requires numerous sampling steps, which is too slow for low-latency robotics; (b) Conditional Flow Matching (CFM) compresses this into a few-step ODE but assumes Euclidean geometry.

Key Challenge: The valid states of SSPs are not arbitrary points in Euclidean space but are constrained to a Clifford torus \(\subset \mathbb{S}^{d-1}\)—every Fourier component \(e^{i\langle\theta_j,x\rangle}\) must maintain unit magnitude. CFM’s linear interpolation \(\phi_t=(1-t)\phi_0+t\phi_1\) corresponds to a chord between two points on a sphere rather than a geodesic. At intermediate times, \(\|\phi_t\|<1\), causing amplitude collapse and phase destruction. Empirical tests show that Euclidean CFM under high noise produces states that "look like valid SSPs" but have shifted spatial positions (Figure 4b) because the phases were corrupted.

Goal: (i) Provide a cleanup method for high-dimensional structured representations like SSPs where "tori are nested in hyperspheres"; (ii) use few-step ODE inference instead of diffusion iterations; (iii) validate cleanup value in a real-world closed-loop spiking neural SLAM system.

Key Insight: Although Chen & Lipman (2024) generalized flow matching to Riemannian Manifolds (RFM), it has only been tested in low dimensions with Gaussian priors. This paper wagers that for extremely high-dimensional, non-Gaussian scenarios (\(d>1000\)) where the target is a gridded SSP encoding, the benefits of geometric priors will stabilize and grow with dimensionality.

Core Idea: Replace the CFM interpolation \(\phi_t = (1-t)\phi_0 + t\phi_1\) with a spherical geodesic \(\phi_t = \mathrm{Exp}_{\phi_0}(t\cdot \mathrm{Log}_{\phi_0}(\phi_1))\). The velocity field \(v_\theta\) regresses only tangent space vectors, ensuring the entire sampling trajectory remains on \(\mathbb{S}^{d-1}\), fundamentally preserving the phase structure.

Method¶

Overall Architecture¶

GFM treats cleanup as a generative transport problem from a noise distribution \(p_0\) to a valid SSP distribution \(p_1\), forcing the transport trajectory to remain on the unit hypersphere \(\mathbb{S}^{d-1}\) rather than performing a discrete search for the nearest prototype.

The pipeline is as follows: The input is a vector \(\tilde\phi\in\mathbb{R}^d\) contaminated by cross-talk, phase drift, or spiking noise, approximating \(p_0\)—an isotropic Gaussian on the hypersphere (sampled via \(z\sim\mathcal{N}(0,I_d)\) and normalized to \(\phi_0=z/\|z\|\)). During training, for each time step \(t\sim\mathcal{U}[0,1]\), \(\phi_0\) is interpolated to a clean SSP \(\phi_1\) along a spherical geodesic to obtain the intermediate state \(\phi_t\). A residual MLP \(v_\theta(\phi_t,t)\) then regresses the tangent space velocity at this point. During inference, a noisy starting point \(\phi_0\) is sampled from \(p_0\), and a \(K\)-step ODE integration based on the Exp-map is performed: \(\phi_{k+1}=\mathrm{Exp}_{\phi_k}(\Delta t\,v_k)\), with explicit normalization at each step to handle numerical drift. The final output \(\phi_K\) is a clean SSP on \(\mathbb{S}^{d-1}\) that can be decoded into spatial coordinates.

Downstream, this cleanup module serves as an online stabilizer between the spiking path integrator (PI) and the VSA map, pulling drifted PI states back to the manifold and stabilizing the SLAM loop.

graph TD
    A["Contaminated SSP Vector φ̃<br/>cross-talk / phase drift / spiking noise"] --> B
    subgraph GEO["Geodesic Interpolation + Tangent Velocity Regression (Training)"]
        direction TB
        B["Spherical Geodesic Interpolation φ_t<br/>Exp/Log mapping, constant unit norm"] --> C["Residual MLP v_θ regresses tangent velocity<br/>cosine flow loss aligns direction only"]
    end
    GEO --> D
    subgraph INF["Geometrically Consistent Inference ODE"]
        direction TB
        D["Sample noisy starting point φ_0 from p0"] --> E["K-step Exp-map Integration<br/>φ_{k+1}=Exp_φk(Δt·v_k)"] --> F["Explicit normalization per step"]
    end
    INF --> G["Clean SSP φ_K (on hypersphere)"]
    G --> H["Downstream: Spiking SLAM online stabilizer<br/>Pull drifted path integrator states back to manifold"]

Key Designs¶

1. Geometry Failure Diagnosis: Proving Euclidean CFM fails in the SSP domain

The work starts by diagnosing why standard CFM fails for SSPs. Noise sources are formalized into three distributions: cross-talk from bundling \(\epsilon\sim\mathcal{N}(0,\tfrac{n-1}{d}I_d)\), phase drift from recurrence \(\mathrm{WrappedNormal}(0,t\sigma^2)\), and decoding noise \(\sigma^2 d/N_{tot}\). The issue is that CFM’s target velocity \(u_t=\phi_1-\phi_0\) corresponds to a chord rather than a geodesic. Chord interpolation results in \(\|\phi_t\|<1\) for \(t\in(0,1)\), which destroys the unit magnitude necessary for Fourier phases \(e^{i\langle\theta_j,x\rangle}\). Visualizations of SSP similarity kernels (Figure 4b) show that Euclidean-generated vectors are shifted in the spatial domain even if they look like valid SSPs—a chain reaction of "amplitude collapse → phase drift → incorrect decoding."

2. Geodesic Interpolation + Tangent Velocity Regression: Ensuring training targets are valid tangent vectors

To maintain the phase structure, GFM moves the probability path to the sphere using Log/Exp maps. Training interpolation becomes \(\phi_t=\mathrm{Exp}_{\phi_0}(t\cdot\mathrm{Log}_{\phi_0}(\phi_1))\), the spherical arc connecting \(\phi_0\) and \(\phi_1\), ensuring \(\|\phi_t\|=1\). The target velocity \(u_t=\frac{d}{dt}\mathrm{Exp}_{\phi_0}(tv)\big|_{v=\mathrm{Log}_{\phi_0}(\phi_1)}\) is the instantaneous tangent along this arc. The loss function is switched from MSE to a cosine flow loss:

\[\mathcal{L}_{\cos}=1-\frac{v_\theta^\top \dot\phi_t}{\|v_\theta\|\,\|\dot\phi_t\|},\]

penalizing only directional deviation, as SSP semantics are encoded in angles. Numerical stability is handled by clipping inner products and adding a floor \(\epsilon=10^{-8}\) to orthogonal components.

3. Geometrically Consistent Inference ODE: Preserving geometry from training to sampling

If inference reverts to Euclidean steps, the geometric priors are lost. GFM performs inference starting from \(\phi_0\sim p_0\) using \(\phi_{k+1}=\mathrm{Exp}_{\phi_k}(\Delta t\,v_\theta(\phi_k,t_k))\), followed by explicit normalization. This requires only a few dozen steps \(K\), making it much cheaper than diffusion. Crucially, the Exp-map update acts as a continuous attractor field rather than a discrete projection, allowing the cleanup module to be inserted into recursive loops without breaking integration dynamics.

Key Experimental Results¶

Main Results¶

Core SLAM results (Table 1, RMSE in meters):

PI Neurons	Method	RMSE (m)	Note
1000	Grid	0.586 ± 0.121	Insufficient resolution
1000	Euclidean FM	0.449 ± 0.068	Linear interpolation
1000	Geodesic FM	0.162 ± 0.055	72% better than Grid
1500	Grid	0.249 ± 0.239	High variance
1500	Euclidean FM	0.204 ± 0.103
1500	Geodesic FM	0.076 ± 0.026	72% better than Grid
2500	Grid	0.083 ± 0.017	Grid catches up with more neurons
2500	Geodesic FM	0.078 ± 0.009

Ablation Study¶

Configuration	Key Metric	Note
Geodesic Flow (full)	0.076m RMSE (1500 neurons)	Full model
Euclidean Flow	0.204m RMSE (1500 neurons)	Replacing geodesic with linear interpolation increases error by ~2.7×
Feedforward Regression	Diffuse spatial distribution	Without ODE iterations, regression collapses to the target mean
Grid Lookup	Fails in SLAM loops	Discrete snapping creates discontinuities that break PI dynamics

Key Findings¶

Geodesics vs. Chords: Upgrading Euclidean FM to GFM reduces RMSE from 0.204m to 0.076m. Figure 4 confirms the "amplitude collapse → phase destruction" causal chain.
Brittleness of Discrete Methods: Grid lookup is strong in static benchmarks but fails in recursive SLAM loops because "snapping" creates jumps that destabilize trajectory integration. GFM provides a smooth continuous attractor field.
Resource Equivalence: GFM with 1500 neurons \(\approx\) baseline with 2500 neurons, effectively saving 40% of neural resources through geometric priors.

Highlights & Insights¶

The "diagnosis-driven" narrative is compelling: formalizing noise sources (3.3) and proving CFM’s failure (3.4) makes the proposed solution a logical necessity.
Redefining cleanup as generative transport is a paradigm shift from discrete searching (Hopfield/Grid) to continuous flow in embedding space. This is applicable to any hyperspherical embedding (e.g., Hyperspherical VAE).
Using cosine flow loss instead of MSE is a critical trick for hyperspherical representations where semantics are angular.

Limitations & Future Work¶

The current \(v_\theta\) is a standard MLP; conversion to a fully spiking network (SNN) for neuromorphic hardware is needed.
The framework is currently tied to hyperspherical topology; other VSA families (e.g., Boolean hypertubes \(\{-1,+1\}^d\)) would require different Log/Exp maps.
SLAM experiments used synthetic 2D benchmarks; testing with real-world odometry or visual front-ends is a future step.

vs. Conditional Flow Matching (Lipman 2022): GFM patches the failure of Euclidean interpolation on curved representation manifolds.
vs. Riemannian Flow Matching (Chen & Lipman 2024): GFM scales RFM to \(d>1000\) and non-Gaussian targets in a real closed-loop system.
vs. Hopfield Networks: GFM treats generative denoising as a continuous attractor, aligning with theoretical views that denoising acts as a continuous associative memory.

Rating¶

Novelty: ⭐⭐⭐⭐ (First scaling of RFM to high-dim structured neuro-symbolic representations).
Experimental Thoroughness: ⭐⭐⭐ (SLAM results are solid, but synthetic datasets and missing trade-off curves for \(K\) are minor gaps).
Writing Quality: ⭐⭐⭐⭐ (Extremely clear causal logic and effective visualizations).
Value: ⭐⭐⭐⭐ (Practical engineering improvements for neuromorphic and neuro-symbolic SLAM).