Finsler-Randers Metric Learning Adds Directionality to Latent-Space Exploration
Summary:This paper introduces a Finsler-Randers metric-learning framework for latent spaces, where standard interpolation implicitly assumes symmetric transition costs. By learning direction-dependent distances, the method can encode irreversible or monotonic transitions that ordinary Riemannian geometry cannot represent directly. Experiments on controlled cross-sectional datasets indicate potential improvements in latent interpolation and extrapolation. The approach is particularly relevant to medical
Research Overview
Deep networks commonly encode images into latent spaces and then treat those representations as ordinary vector sets without an explicit geometry. This works well for classification and retrieval, but it becomes problematic when the system must interpolate between states or extrapolate to unseen states. In that setting, the meaning of distance in the latent space should reflect the cost of a real-world transition.
The paper identifies asymmetry as a central limitation. Conventional latent interpolation assumes symmetric transition costs:
[ d(x,y)=d(y,x) ]
That assumption is inappropriate for many irreversible processes. Disease progression may allow movement from an earlier stage to a later one while making the reverse transition implausible. A symmetric straight-line interpolation can therefore generate intermediate states that violate the underlying dynamics.
From Riemannian to Finsler-Randers Geometry
Riemannian geometry introduces local curvature through a smoothly varying inner product, but its induced distance remains symmetric. The proposed approach uses a Finsler-Randers structure:
[ F(x,v)=\sqrt{\alpha(x,v)^2+\beta(x,v)^2} ]
Here, (F(x,v)) is the local cost of moving at (x) in direction (v). The base length term (\alpha) defines local scale, while the Randers term (\beta) encodes direction-dependent asymmetry. Unlike a symmetric distance, the Finsler length may satisfy:
[ F(x,v)\neq F(x,-v) ]
Thus, moving from one state to another may cost more than moving back. The framework learns these directional parameters from data rather than requiring users to manually specify disease stages or temporal direction.
Mechanism and Monotone Trajectory Constraints
The method is designed to produce latent trajectories that respect monotonic sequential constraints. For samples ordered by clinical time or progression stage, the optimization can favor paths that move through states in the permitted direction.
In local coordinates, the Randers component is commonly expressed as:
[ \beta(x,v)=b(x)^T v ]
The vector field (b(x)) varies with position. It acts as a learned local bias that distinguishes forward and backward movement. Integrating this term along a path produces path-dependent costs, allowing different routes between the same endpoints to carry different physical meaning.
Training must jointly model local scale, the direction field, and sequential constraints. Optimizing endpoint distances alone may produce a useful retrieval embedding but still generate trajectories that violate monotonicity. Evaluation should therefore measure both geometric fit and directional feasibility.
Engineering Trade-offs
The expressive power of asymmetry comes with additional optimization and validation costs.
First, learning a non-symmetric metric is more difficult than fitting a conventional Euclidean or Riemannian model. The system must estimate a directional field at each latent location while handling unequal sample coverage across directions. If training data are dominated by forward transitions, the reverse direction may be poorly constrained.
Second, a learned direction does not automatically carry causal meaning. If labels, disease stages, or acquisition order contain bias, the model may encode that bias as a valid progression direction. Clinical deployment would require validation across hospitals, scanners, and patient populations.
Third, path computation is more expensive than linear interpolation. Numerical integration introduces step-size sensitivity, while gradient-based trajectory optimization can add substantial latency. A practical system should retain inexpensive interpolation as a fallback and invoke constrained geometric paths only for samples that need them.
Finally, the approach depends on the quality of the underlying representation. If an encoder maps different stages to disconnected regions with no meaningful continuity, metric learning can describe relationships between clusters but cannot create clinically valid intermediate states. The geometry should ideally be trained jointly with temporal consistency and interpolation fidelity.
Evaluation Considerations
The paper demonstrates potential for interpolation and extrapolation on controlled cross-sectional datasets. However, the available abstract does not report dataset size, encoder architecture, optimization protocol, error metrics, or statistical significance, so the real-world magnitude of the benefit remains unresolved.
A robust comparison should include Euclidean interpolation, Riemannian interpolation, a symmetric Finsler variant, and the full directional Finsler-Randers method. All variants should use the same encoder and data split to isolate the contribution of geometry.
Relevant metrics include:
- Interpolation and extrapolation errors;
- Violation rates for predefined monotone constraints;
- Correct ranking of valid versus invalid transition directions;
- Generalization across sites, devices, and patient cohorts;
- Trajectory-optimization latency and memory overhead.
Visual smoothness alone is insufficient. A convincing biomedical evaluation must verify that the learned geometry distinguishes permissible transitions from forbidden or irreversible ones.
Deployment Guidance
For medical-imaging workflows, the method can initially be attached to a frozen pretrained encoder. This isolates the geometric hypothesis before jointly fine-tuning the encoder and metric parameters. Patient-, time-, and site-level splits should prevent leakage from adjacent slices or repeated examinations.
For generative models, the framework is most relevant when states evolve directionally, such as pathological change, equipment degradation, or materials evolution. It is less suitable for data without reliable ordering, because the system cannot infer which direction represents progression.
A practical deployment can expose three tiers: Euclidean interpolation for batch previews and fallback, Riemannian or symmetric Finsler methods for routine smoothing, and the full directional model for constrained high-risk cases. Ablation studies should quantify the incremental value of asymmetry before attributing improvements to the metric framework itself.
Conclusion
This work shifts the discussion of latent-space exploration from whether a space is curved to whether transitions are directional. Finsler-Randers metric learning offers a more expressive representation for monotonic and irreversible processes than ordinary Riemannian geometry. The motivation is technically well grounded, but the current evidence comes primarily from controlled cross-sectional experiments. Clinical or production use requires stricter cross-dataset validation, temporal-leakage controls, and explicit assessment of trajectory-optimization costs.
— END —Source: ArXiv Machine Learning (cs.LG) (2026-11-05)
Tags: #Finsler Geometry #Randers Metric #Latent Space #Asymmetric Distance #Metric Learning
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