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Robust blind unmixing: A geometric approach to overcoming basis variation

Dumitru Mirauta, Vladimir V. Gusev, Michael W. Gaultois, Matthew J. Rosseinsky, Yannis Goulermas

arXiv:2610.04091v1cs.LGcs.AI

TL;DR

Basis variation means mixtures of the same underlying objects can differ, complicating recovery of their component patterns. The paper uses a geometric metric-space approach with optimal-transport tools to handle such variation. Across random GMM, simulated PXRD, and laboratory HI data, it reports outperforming other general-purpose methods in a range of unmixing scenarios.

  • Problem

    The same underlying object can yield different patterns across mixtures, making recovery of basis patterns from mixtures a challenge.

    Source: §1 · p. 2

  • Method

    The approach uses metric-space geometry and Wasserstein tools, including sparse signatures and efficient 1D distances, transport maps, and centroid calculations.

    Source: §6 · p. 31

  • Results

    Across random GMM, simulated PXRD, and laboratory HI data, the approach consistently outperforms other general-purpose methods in a range of unmixing scenarios.

    Source: §6 · pp. 31–32

  • Takeaways & Limitations

    Sparse signatures, fast 1D transport routines, and displacement interpolation make the method computationally feasible for exploratory data analysis.

    Source: §6 · p. 31

    Novel patterns that act as outliers may be drowned out by more common bases, so detecting them requires a dedicated anomaly-detection approach.

    Source: §3.4 · p. 18

Abstract

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Signal separation problems are common in science. A prominent example of this occurs during the use of diffraction or spectroscopy to identify the individual components of a mixture by measuring it. In the simplest case, the measured signal is a linear combination of basis patterns corresponding to the constituent parts. The unmixing problem is to infer all or some of these basis patterns and abundances of components from measurements of distinct mixtures. One of the core challenges of this task is the variation of the basis from mixture to mixture due to noise and the exact physics of the measurement process. This is usually addressed with tailored model-based and parametric methods that are then limited in use to specific application domains by the nature of the assumptions made. We propose a novel geometric approach to unmixing problems which views the generation of data during measurement through a metric space lens, thereby shifting the focus from parametrised models to a general relationship between basis transformations and the corresponding geometry. We take advantage of the optimal transport distances to capture commonly occurring basis variations, and use minimisation of in-class variance of candidate solutions to drive the optimisation. We pay special attention to the one-dimensional case due to its practical importance and availability of efficient distance and transport map routines. The effectiveness of our approach is demonstrated on a range of unmixing tasks using random Gaussian mixture models, simulated powder X-ray diffraction, and laboratory hyperspectral imaging datasets.

1 Introduction

The introduction frames unmixing as recovering constituent patterns from mixtures when one underlying object can produce multiple patterns. It motivates a geometric, metric-space approach that accommodates basis variation without relying on a fixed parametric model.

1 Introduction

  • Basis variation means one underlying object may produce distinct patterns, and mixtures combine varying basis patterns in proportions that can yield different measurements.The paper treats patterns as normalized distributions and defines a mixing model through both basis-variation and combination mechanisms.Source: §1 · p. 2
  • Blind unmixing seeks to recover a basis from mixtures alone, while variation across mixtures can reveal different basis parts that are obscured in others.Complex variation can also make even nonblind inversion difficult because a mixture may admit many explanations.Source: §1 · p. 2
  • Tailored models may not capture the range of transformations across applications, while a broadly applicable method could reduce manual analysis in materials discovery.The introduction notes that many characterization techniques are used and expert time can otherwise go to manual analysis rather than interpretation and experiment planning.Source: §1 · pp. 2–3
  • For transformations Lipschitz in a chosen metric space, basis variations are expected to form separable clusters up to a limit that depends on true-basis distinctiveness.The resulting patterns can be represented in different spaces, including vectors, Gaussian mixture parameters, or abstract measures.Source: §1 · p. 3
  • Wasserstein distances accommodate distribution-like data and transformations such as shifts that can appear discontinuous on a shared uniform grid.The introduction identifies shifting and broadening as transformations for which Wasserstein distances are Lipschitz.Source: §1 · p. 3
  • The approach allows non-Euclidean perturbations without fitting a fixed basis-variation model and focuses on decomposing a given mixture rather than constructing a similar one..Source: §1 · p. 3
  • The method uses mixing geometry to guide plausible decompositions rather than relying on a regressive cost..Source: §1 · p. 3
  • Sparse, dynamic signatures can represent Wasserstein centroids more accurately and enable efficient separation..Source: §1 · p. 3

2 Preliminaries

The preliminaries establish index conventions and an implicit measurability convention for subsets.

2 Preliminaries

  • Lower indices denote vector indices..Source: §2 · p. 4
  • Upper indices may enumerate non-vector quantities, such as discrete measures..Source: §2 · p. 4
  • A subset notation X ⊆ X implicitly refers to measurable subsets..Source: §2 · p. 4

Acronyms

The section establishes a flexible mixing model in which bases may vary before combination, then introduces transport and geometric perspectives for framing unmixing.

Acronyms

  • The acronym key expands core terms including OT, EMD, GMM, HI, NMF, PXRD, and centered Wasserstein variants.Source: Acronyms · p. 5

2.1 Formal definition of mixing and unmixing

  • Mixing is modeled as κ(aj; ϕ(Λj; B)), and blind unmixing seeks to recover B from the mixtures alone.Source: §2.1 · pp. 5–6

2.2 Optimal transport primer

  • Optimal transport finds minimum-cost associations between distributions, with transport maps describing mass redistribution and the earth-movers analogy interpreting cost as movement.Source: §2.2 · pp. 6–7

2.2.1 Discrete optimal transport

  • Finite measures are represented by signatures containing support positions and weights, which restrict discrete transport plans to pairs of support points.Source: §2.2.1 · p. 9

2.3 Existing foundational techniques

  • Linear unmixing becomes C = BA, but reconstruction fitting can reward implausible bases under flexible variation and pointwise costs can create avoidable local minima.Source: §2.3 · pp. 9–10

2.3.1 Geometric approaches

  • With dense sampling of linear mixtures, basis recovery reduces to finding convex-hull vertices; manifold analogues extend this geometry but may not model independent basis variation.Source: §2.3.1 · p. 11

3 Latent basis variant clustering

The method searches among exact mixture reconstructions for basis variants that cluster, using Fréchet variance to favor solutions requiring less average modification. It alternates Wasserstein-based averaging and separation, with a specific boundary for detecting novel patterns.

3 Latent basis variant clustering

  • Minimizing Fréchet variance among exact reconstructions favors clustered basis variants that require less average modification to explain the data.The paper presents these bases as a more natural explanation and uses Fréchet variance to measure variation.Source: §3 · p. 12

3.1 Averaging

  • Wasserstein centroids average patterns along geodesics, and a hierarchical approximation is used when averaging more than two patterns.Displacement interpolation moves mass continuously along ground-space geodesics; hierarchical approximation follows Boissard et al. (2015).Source: §3.1 · pp. 13–14

3.2 Separation

  • Separation first retrieves abundances, then targets transformed basis patterns rather than stopping at a linear representation.The combination constraint also lets the separation subproblem minimize with respect to weights alone.Source: §3.2 · pp. 16–17

3.3 Proposed unmixing algorithm

  • The proposed algorithm alternates basis averaging and mixture separation, with independent subproblems enabling parallel computation when plans and centroids can be computed efficiently.The paper gives a concrete implementation for discrete measures with the W2 distance.Source: §3.3 · p. 18

3.4 Variations of the unmixing problem

  • Partially blind unmixing can hold known patterns fixed, but novel patterns may be drowned out as outliers and require dedicated anomaly detection.The paper describes residual-based detection as an option for mixtures where anomalies are reasonably suspected.Source: §3.4 · pp. 18–19

4 Fast Wasserstein

For one-dimensional transport, the north-west corner algorithm gives an efficient construction and, under a stated cost condition, an optimal map. The section also develops quantised proxies for approximating Wasserstein distances with fewer support points.

4 Fast Wasserstein

  • The section develops both fast exact and approximate ways to exploit structure in optimal-transport calculations.Quantised measures provide the approximate-distance route; the north-west corner algorithm addresses the exact 1D case.Source: §4 · p. 19, §4.1 · p. 19, §4.2 · p. 22

4.1 The north west corner algorithm

  • The north-west corner algorithm constructs the monotone transport plan in O(N) time for 1D Euclidean ground distance, without minimisation.In 1D, monotonicity fully constrains transport plans; the algorithm constructs the unique feasible map for ground costs admitting monotone increasing solutions.Source: §4.1 · p. 19, Figure 9 · p. 21

4.1.1 Optimality for 1D sorted measures

  • In 1D, the north-west corner algorithm on sorted signatures produces an optimal map when the ground cost satisfies the stated inequality.The algorithm constructs the unique monotone feasible map in O(N) time for ground costs admitting monotone increasing solutions, including Euclidean distance.Source: §4.1 · p. 19, §4.1.1 · p. 22

4.2 Optimal distance approximation quantisers

  • Hierarchical centroid-linkage proxies improve monotonically with quantisation accuracy, and can represent distributions with half as many Diracs as uniform-grid downsampling.The paper also notes that halving marginal sizes yields transport plans a quarter the size, with potential solve-time speedups of 16–64 under the stated naive linear-program scaling.Source: §4.2 · pp. 22, 24

5 Experimental evaluation

The evaluation compares the approach with NMF and oNMF across random GMM, simulated PXRD, and laboratory hyperspectral data. The results show lower PXRD abundance error and distinguish performance across linear and more complex mixing scenarios.

5 Experimental evaluation

  • The evaluation covers 1,200 random GMM datasets with varying basis-variation strengths, a physically informed PXRD simulation, and three laboratory hyperspectral scenes.The PXRD task uses non-blind separation because none of the compared methods consistently discovered all six basis patterns blindly.Source: §5 · p. 24, §5.2 · p. 28

5.1 Random GMM datasets

  • On random GMM datasets, oNMF sometimes extracts constituents less accurately than NMF despite improving overall fit, illustrating the risk of overly flexible composition models.Figure 13 depicts mixtures whose non-uniform peak shifts require independent peak perturbations to explain them with a shared basis.Source: §5.1 · pp. 26, 28, Figure 13 · p. 27

5.2 PXRD unmixing

  • Table 4 reports significantly lower abundance-prediction error for the approach than the compared methods on the synthetic PXRD dataset.The comparison is in the separation-only, fixed-basis regime.Source: §5.2 · p. 28, Table 4 · p. 29

5.3 Hyperspectral imaging

  • In hyperspectral scene 1, the method avoids unnecessary basis variation and gives decompositions similar to the appropriately constrained linear model; it improves somewhat on linear decomposition in scene 2 and greatly in scene 3.Scene 2 includes second-order reflections among powders, while scene 3 contains second-order reflections between the board and powders.Source: §5.3 · pp. 29, 31

6 Conclusion

The paper frames unmixing geometrically, using feasible latent point clouds and a Fréchet-variance cost to favor solutions with less basis variation. Experiments across GMM, PXRD, and hyperspectral data show the approach outperforming other general-purpose methods, while higher-dimensional applications remain future work.

6 Conclusion

  • The proposed in-class Fréchet-variance cost searches for latent point clouds that explain measurements while requiring the least basis variation.For Lipschitz variation functions, the paper expects clustering in the metric space, with tighter clusters under smaller bounds on variation.Source: §6 · p. 31
  • For shifting and broadening, the implementation uses Wasserstein distance and optimal-transport features to make one-dimensional calculations computationally feasible.It uses sparse representative signatures, a linear-time exact 1D Wasserstein and transport-map algorithm, and displacement interpolation for centroids.Source: §6 · p. 31
  • Across random GMM, simulated PXRD, and laboratory hyperspectral data, the approach consistently outperforms other general-purpose methods across a range of unmixing scenarios.Source: §6 · pp. 31–32
  • The paper identifies higher-dimensional data as a future direction that requires fast nD transport-map calculation.Other proposed directions include clustering with between-class variance, bidirectional partial matching for anomaly detection, and alternative ground distances.Source: §6 · p. 32

Declarations

The authors acknowledge support from Johnson Matthey, the Leverhulme Trust, and EPSRC; V. V. Gusev received a Leverhulme Research Fellowship.

Declarations

  • The PhD research was funded by Johnson Matthey, with additional Leverhulme Trust and EPSRC support; V. V. Gusev held a Leverhulme Research Fellowship.EPSRC support included grants EP/V026887, EP/Y028775/1, and EP/Y028759/1.Source: Declarations · p. 32

Appendix A Clustering and separability of metric space perturbations

The appendix gives geometric conditions under which perturbed basis patterns remain identifiable by Voronoi cells. A Lipschitz variation model yields a parameter bound for separability, while discrete shifting with Euclidean distance can violate it.

Appendix A Clustering and separability of metric space perturbations

  • Clusters remain identifiable when their maximum radius is below R and distinct bases are spaced more than 2R apart.Under these conditions, each cluster lies within its basis pattern’s Voronoi cell.Source: Appendix A · p. 32
  • A Lipschitz variation map guarantees Voronoi separability when the parameter distance from the unvaried basis stays below the minimum basis spacing divided by twice the largest Lipschitz constant.The theorem assumes the variation map equals the original basis at a reference parameter.Source: Appendix A · p. 33
  • For a compact pattern space, the variation radius must be strictly less than half the maximum basis spacing.The appendix notes that this bound is necessary because the maximum spacing is finite.Source: Appendix A · p. 33
  • Discrete shift-down can move a pattern by the maximum Euclidean distance, so shifted bases are not Voronoi separable over the specified hull.The example concerns the hull of the coordinate basis vectors and zero.Source: Appendix A · p. 33

Appendix B Wasserstein quantisation and weighted k-means

The appendix formulates Wasserstein quantisation as a semidiscrete optimal-transport problem and describes an alternating region-and-centroid procedure. Its updates are non-increasing in cost and converge to a local minimiser.

Appendix B Wasserstein quantisation and weighted k-means

  • With regions fixed, optimal discrete support locations are their centroids; with locations fixed, optimal regions are Voronoi cells.Reassigning points to a closer support location improves transport cost when a region violates this assignment rule.Source: Appendix B · pp. 33–34
  • The alternating quantisation updates have non-increasing cost and converge to a local minimiser of the quantisation objective.For fixed regions, the optimal support locations are centroids; for fixed locations, the optimal regions form a Voronoi tessellation.Source: Appendix B · pp. 33–34
  • The alternating procedure is a generalised weighted k-means algorithm, with Lloyd’s algorithm as a special case.The stated special case uses X = R and interval-shaped regions.Source: Appendix B · p. 34

B.1 Wasserstein quantiser monotonicity and convergence

The appendix establishes two properties of optimal Wasserstein quantisers: their error decreases as support size grows, and the quantisers converge weakly to the target measure.

B.1 Wasserstein quantiser monotonicity and convergence

  • The optimal W2 quantisation error strictly decreases when support size increases from N to N+1.A constructed size-(N+1) quantiser has lower cost than the optimal size-N quantiser, and the optimal size-(N+1) cost is no greater than that construction’s cost.Source: §B.1 · p. 34
  • Optimal W2 quantisers with support size N weakly converge to the target measure as N grows.The proof uses a sequence of suboptimal quantisers converging to the measure and compares the optimal quantisers with them.Source: §B.1 · p. 34

Appendix C Weighted centroids parameterise the geodesic

Under a condition that intermediate points can realize specified distances to both endpoints, the weighted centroids lie on the geodesic between the two patterns and traverse it at constant speed.

Appendix C Weighted centroids parameterise the geodesic

  • If intermediate points can realize the required distances to both endpoints, the weighted centroids lie on the geodesic and parameterize it at constant speed.The proof shows the barycentric cost bound is tight for interior weights, giving distances proportional to the weight parameter.Source: Appendix C · p. 35

Appendix D General McCann interpolation

The appendix connects optimal transport trajectories in the ground space to geodesics between distributions in Wasserstein space. Under a compatibility condition on the transport maps, this yields barycentric interpolation, including in one dimension.

Appendix D General McCann interpolation

  • The dynamic formulation evaluates the full trajectory rather than only endpoint displacement, and agrees with the static problem when static cost matches the best trajectory cost.The appendix defines geodesics as minimum-length, constant-speed paths and geodesic spaces as spaces where every pair of points is connected by one.Source: Appendix D · pp. 35–36
  • A constant-speed optimal trajectory consists of ground-space geodesics and can be expressed using ground-space barycenters between each point and its final transport.Where Lemma 5 applies, including in one dimension with unique feasible monotonic plans, these barycenters give Wasserstein barycenters via displacement interpolation.Source: Appendix D · p. 36
  • When the transport maps satisfy the stated compatibility condition, the trajectory of transported measures is a Wasserstein space geodesic between μ and ν.The condition requires the static map between intermediate measures to carry each earlier trajectory point to its later point.Source: Appendix D · p. 36
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