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The Silhouette Operator: Identifiability of Low-Rank Measures from One-Dimensional Projections

Robert A. Vandermeulen

arXiv:2610.09687v1math.STcs.ITcs.LGmath.FAmath.ITstat.MLstat.TH

TL;DR

High-dimensional estimation is difficult, and continuous mixture-of-products density models remain challenging to estimate efficiently. The paper defines silhouette operators for identifying low-rank measures from projected marginals and introduces SME; suitably chosen 2k projections are optimal, and SME achieves the highest mean test log-likelihood on both evaluated datasets.

  • Problem

    Continuous mixture-of-products density estimation remains challenging, while high-dimensional statistical and computational difficulty often grows rapidly with dimension.

    Source: §1 · pp. 1, 3

  • Method

    The paper defines silhouette operators as tuples of 2k projected marginals and fits SME by matching selected one-dimensional projections of pairwise model marginals to empirical projections.

    Source: §1 · p. 2, §5.2 · p. 14

  • Results

    A suitably chosen 2k projections identify rank-≤k measures and are optimal in number; SME achieves the highest mean test log-likelihood on both evaluated datasets.

    Source: §1 · p. 2, §6.2.3 · p. 22

  • Takeaways & Limitations

    In the tested sample-limited, moderate-dimensional regime, SME retains mixture flexibility while reducing within-component estimation to one-dimensional problems.

    Source: §6.2.3 · p. 22

    Projection directions cannot be arbitrary, and tail and boundedness assumptions cannot generally be omitted from the identifiability and homeomorphism results.

    Source: §1 · pp. 2–3

Abstract

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Structured recovery phenomena, such as restricted isometry properties in compressed sensing, have shown that high-dimensional objects can often be reconstructed from remarkably low-dimensional linear measurements. This work develops an analogous recovery framework for low-rank signed measures on $\mathbb{R}^2$, defined here as measures that can be expressed as finite sums of product measures with one-dimensional factors. The framework is based on linear operators, termed "silhouette operators," that map a measure to a fixed finite collection of one-dimensional linear pushforwards. The main results show that a suitably chosen collection of $2k$ projected marginals suffices to identify every compactly supported rank-$\le k$ signed measure, that this number is optimal, and that the projection directions cannot be chosen arbitrarily. The framework is also extended to higher-dimensional sums of product measures by establishing sufficient conditions under which collections of pairwise marginals identify the full model. Building on this framework, a computationally efficient estimator, termed "silhouette mixture estimation" (SME), is introduced for constructing a low-rank empirical measure from data by matching its one-dimensional projected marginals to the corresponding empirical marginals in Wasserstein distance. When combined with one-dimensional density estimators, SME yields an efficient nonparametric density estimator that performs strongly relative to a range of parametric, nonparametric, and deep-learning baselines in settings of moderate dimension and sample size.

1 Introduction

The paper develops a recovery framework for low-rank measures inspired by restricted-isometry phenomena for matrices. It uses finite collections of one-dimensional projections for identification and builds on that framework for estimation and higher-dimensional models.

1 Introduction

  • Rank-≤k measures on R2 are finite sums of at most k product measures, a structure directly analogous to matrix rank.Rank exactly k is equivalent to having k linearly independent factors in each coordinate, as stated in Proposition 13.Source: §1 · pp. 1–2
  • For compactly supported and sub-Gaussian rank-≤k measures, a suitably chosen collection of 2k linear pushforwards uniquely identifies the measure.The tuple of projected marginals is called its silhouette.Source: §1 · p. 2
  • No collection of 2k−1 directions identifies all rank-≤k signed measures, and even 2k directions can fail when their geometry is unsuitable.The silhouette operator is also a weak∗ homeomorphic embedding on rank-≤k probability measures supported in a fixed compact set.Source: §1 · p. 2
  • In higher dimensions, collections of pairwise marginals can identify the full model under assumptions adapted from standard matrix and tensor identifiability conditions.These include linear independence and joint irreducibility.Source: §1 · pp. 2–3
  • SME estimates low-rank empirical measures separately from component-density estimation, keeping choices such as density-estimation bandwidths out of its mixture-fitting objective.Its objective uses univariate Wasserstein distances that can be evaluated rapidly and in parallel.Source: §1 · p. 3
  • SME performs strongly against parametric, nonparametric, and deep-learning baselines, particularly in moderate-dimensional settings with limited sample sizes.The introduction presents density estimation as a natural downstream application of the resulting representations.Source: §1 · p. 3

2 Main Theorems

The section establishes when low-rank measures are identifiable from one-dimensional projections and characterizes topological and geometric properties of the resulting silhouette operators.

2.1 Notation and Definitions

  • The notation section defines pushforwards as linear in the measure and specifies Mexp(R^d), which includes compactly supported and sub-Gaussian measures.These definitions set the measure and tail-condition framework used in the identifiability results.Source: §2.1 · p. 4

2.2 Injectivity of the Silhouette Operator

  • A measure with component marginals in Mexp(R) is determined by its 2k projections along the theorem’s specified directions, including when its rank is below k.The result applies when the component marginals of both measures satisfy the stated condition.Source: §2.2 · p. 4
  • Fewer than 2k directions cannot identify every rank-≤k measure; symmetric direction configurations can fail, and signed-measure identifiability depends on component-tail conditions.Coordinate-wise transformations into a compact set preserve the sum-of-products structure and can address tail issues.Source: §2.2 · p. 5

2.3 Topological and Geometric Properties of the Silhouette Operator

  • For compact supports, silhouette operators are weak∗ homeomorphisms on total-variation-bounded subsets of rank-≤k measures.The boundedness condition cannot generally be dropped.Source: §2.3 · p. 6
  • For probability measures in the stated compactly supported rank-≤k class, convergence in distribution of each projected marginal implies convergence in distribution of the measures.This follows from the weak∗ topological result and the correspondence between weak∗ convergence and convergence in distribution for probability measures.Source: §2.3 · p. 6
  • At the population level, injectivity makes the target the unique rank-≤k minimizer of any discrepancy that vanishes exactly when silhouettes agree.Replacing the target silhouette with the empirical silhouette yields an estimation strategy based on one-dimensional projections, whose direction-specific computations can be parallelized.Source: §2.3 · p. 6
  • A Banach-space tensor-product result establishes the weak∗ closedness of rank-≤k measures used in the proof of the topological theorem.The authors note that this structural result may be of independent interest.Source: §2.3 · p. 6
  • On the intersection of any finite-dimensional subspace with the rank-≤k class, rescaled silhouette operators preserve distances up to a uniform distortion factor.With finite-dimensional component spaces, this inequality takes the form of the classical low-rank matrix RIP.Source: §2.3 · pp. 6–7

3 Higher-Dimensional Extensions of the Silhouette Framework

The section extends silhouette-based identification to higher-dimensional mixture-of-products models using selected pairwise marginals. Recovery depends on the marginal graph and structural conditions on component measures.

3 Higher-Dimensional Extensions of the Silhouette Framework

  • The extension treats higher-dimensional sums of product measures and seeks conditions for recovering them from collections of pairwise marginals.Each included pairwise marginal can itself be identified through its silhouette operator.Source: §3 · pp. 7–8
  • A connected marginal graph is necessary outside trivial cases; estimating its pairwise marginals ranges from d −1 edges to Θ(d^2), affecting computational cost.A disconnected graph leaves coordinate groups uncoupled, so the full model cannot generally be identified.Source: §3 · p. 8
  • With linearly independent true component marginals along each coordinate, identifying components along one coordinate suffices for recovery from any connected graph.The independence condition applies to the true model, not as a constraint during estimation; the recovered components and weights agree up to permutation.Source: §3 · p. 8
  • Theorem 5 does not explain how to obtain the initial coordinate components, while the stated identifiability conditions are not exhaustive.For example, the initial components may already be known or estimated by another method; further guarantees may arise from interactions between component structure and graph connectivity.Source: §3 · pp. 8–9
  • Joint irreducibility of component families on one coordinate pair can identify the components from that pairwise marginal, then propagate recovery through the connected graph.Theorem 5 provides the propagation step once a coordinate's components are identified.Source: §3 · pp. 8–9
  • Under singularity on one coordinate pair, Theorem 7 can identify a single component and propagate it when that pair is observed and every other coordinate neighbors one of its endpoints.The identified component must be separated from the others along the specified pair; candidate factorizations need not be constrained to contain it.Source: §3 · p. 9

4 Related Work

Related work places the paper alongside nonparametric mixture models, compressed-sensing recovery, and projection-based methods. These connections highlight both the model’s structure and the role of finite projections.

4.1 Mixture-of-Products Models

  • Mixture-of-products models represent covariates as conditionally independent given a latent discrete variable, and prior work studies their identifiability, estimation, and statistical rates.For Lipschitz component densities, prior work reports an eO(n−1/3) L1 rate [13]; other methods use matrix or tensor factorization [13] [16] [17] [30] [31] [32].Source: §4.1 · pp. 9–10

4.2 Restricted Isometry and Low-Rank Matrix Recovery

  • The silhouette operator is a deterministic restricted-isometry-type construction for low-rank measures, unlike classical RIP guarantees developed through random measurements.Classical random constructions use concentration and covering arguments, whereas the silhouette result uses a fundamentally different proof approach.Source: §4.2 · p. 11

4.3 Other Related Topics

  • Unlike finite-parameter elliptical recovery, 2k silhouette projections recover measure-valued component marginals for rank-≤k measures.The paper also connects projections to sliced Wasserstein methods, whose one-dimensional computations reduce the cost relative to multidimensional optimal transport.Source: §4.3 · p. 12
  • Pairwise silhouettes also relate to deconvolution and graphical models, but the marginal graph identifies latent mixture structure rather than encoding conditional independence.Using sparse projection directions limits each component to a two-factor convolution; in tree- or forest-structured graphical models, edge marginals instead encode conditional independence relations.Source: §4.3 · pp. 12–13

5 Algorithm and Applications

SME fits a mixture-of-products empirical measure by matching projected one-dimensional marginals, then uses its blockwise product structure for efficient computation and downstream tasks. The section describes the objective, optimization, and ways to use the fitted estimator.

5.1 Form of the Estimator

  • The estimator mixes block-level product measures weighted by block size, and its coordinate marginals exactly match those of the empirical distribution for every partition.Each block uses empirical marginals for its coordinates; the subsequent choice of partition specifies the estimator.Source: §5 · p. 13, §5.1 · p. 14

5.2 Objective Function

  • The fitting objective aggregates one-dimensional projection discrepancies between candidate and empirical pairwise marginals across selected coordinate pairs and directions.The experiments use 1-Wasserstein distance; selected projection directions and graph edges determine which comparisons are included.Source: §5.2 · pp. 14–15

5.3 Computation of Projected Marginals

  • Projected SME marginals are computed blockwise and mixed using block-size weights, then compared with empirical projections to form the fitting objective.For each coordinate pair and direction, the projected candidate and empirical marginals are compared using the chosen metric.Source: §5.3 · p. 15

5.4 Optimization

  • SME selects a data partition by minimizing the aggregate projection discrepancy, using objective structure for practical optimization despite the exponential assignment space.Terms across edges and directions can be evaluated independently; caching empirical and block contributions makes local reassignment updates cheaper.Source: §5.4 · p. 16

5.5 Utilizing the SME Estimator

  • The mixture-of-products representation supports efficient sampling and density estimation, while conditioning changes mixture weights but leaves component marginals on unobserved coordinates unchanged.For ℓ∞-ball probabilities, coordinate-wise interval probabilities can be multiplied within blocks and combined by block weights, avoiding explicit enumeration of combinatorial support.Source: §5.5 · pp. 17–18

6 Experiments

Experiments evaluate SME on synthetic mixture recovery and real-world density estimation. SME achieves the highest mean test log-likelihood on both tested datasets, while synthetic recovery improves substantially when a second non-axis projection is added.

6.1 Mixture Recovery on a Synthetic Dataset

  • The synthetic test uses an identifiable three-component mixture whose components have overlapping supports but retain regions unique to each component.The model satisfies the joint irreducibility assumption, and the ground-truth panel contains 900 samples, 300 per component.Source: §6.1 · pp. 18–19
  • Adding a second non-axis projection substantially reduces directional leakage, while four- and six-projection fits still show some overlap-region errors and isolated misassignments.The figure shows ground-truth and recovered component assignments for fits using 1, 2, 4, and 6 non-axis projections.Source: Figure 2 · p. 19, §6.1 · p. 20

6.2 Density Estimation on Real-World Datasets

  • Real-world density estimation is evaluated on HEPMASS and kin8nm using mean test log-likelihood over 20 random train-validation-test splits.Coordinates are standardized, and the reported sample-limited regime is distinguished from the much larger HEPMASS training setting in [80].Source: §6.2 · p. 20, §6.2.3 · p. 22

6.2.1 Density Estimation Methods

  • The SME density estimator combines component-coordinate one-dimensional Gaussian kernel density estimates after selecting a partition and number of mixture components.The component count is selected from 2, 3, or 4 by validation likelihood; bandwidths are selected by 15-fold cross-validation.Source: §6.2.1 · p. 20

6.2.2 Choice of Baselines

  • Baselines span factorized KDE extremes, axis-aligned and full-covariance Gaussian mixtures, and a masked autoregressive flow.The comparisons represent differing structural assumptions, with MAF included as a deep density-estimation baseline.Source: §6.2.1 · pp. 20–21, §6.2.2 · p. 21

6.2.3 Results and Discussion

  • SME achieves the highest mean test log-likelihood on HEPMASS and kin8nm in the sample-limited, moderate-dimensional experimental settings.The table reports test log-likelihood across 20 matched random splits; the evaluation differs from the large-sample HEPMASS flow setting in [80].Source: Table 1 · p. 21, §6.2.3 · p. 22

7 Sharpness of the Main Theorems

The section establishes that the 2k projection count is sharp, but identifiability also depends on direction geometry and tail assumptions. A uniform total-variation bound is needed for the stated homeomorphic embedding result.

7. Sharpness of the Main Theorems

  • No collection of 2k−1 directions identifies all rank-≤k probability measures, even when component marginals have smooth, compactly supported densities.Thus the lower bound is not specific to signed measures, irregular densities, or heavy tails.Source: §7 · p. 22
  • Even 2k directions can fail when arranged in certain symmetric configurations, making projection geometry part of the identifiability condition.The results identify reflection symmetry among directions as a potential obstruction.Source: §7 · p. 23
  • For k ≥ 2, finite-projection identifiability can fail with component densities that decay faster than any polynomial and have moments of all orders.Theorem 1 instead assumes component Laplace transforms are finite on a neighborhood of the origin.Source: §7 · pp. 23–24
  • Without a uniform total-variation bound, the silhouette map need not be a weak∗–weak∗ homeomorphic embedding, even for smooth compactly supported rank-one measures.A constructed sequence has diverging total variation norms while every prescribed projection converges to zero in total variation.Source: §7 · p. 24

8 Directions for Future Research

The paper identifies open directions for quantitative stability of the silhouette operator and for using silhouettes in statistical tasks beyond density fitting.

8. Directions for Future Research

  • Future work could seek stability estimates and determine whether extra projections improve them after identifiability has been achieved.Such guarantees may depend on the measure class, projection directions, and metrics on the domain and range.Source: §8 · p. 24
  • Finite silhouettes could support two-sample tests because, under the identifiability assumptions, equality of the projected laws implies equality of the original distributions.The paper leaves open which other statistical tasks can be recast in silhouette space and what guarantees or efficiencies would follow.Source: §8 · p. 24

9 Proof of Injectivity of the Silhouette Operator on Low-Rank Measures

The section proves injectivity by characterizing minimal product-measure rank and developing Laplace-transform tools for measures and finite mixtures. It uses rank criteria and convolution properties to rule out a considered failure case, then applies Theorem 1 to conclude equality of the measures.

9.1 Signed Measure Preliminaries and Rank

  • A product-measure decomposition has rank k exactly when its component measures are linearly independent in each coordinate.If either coordinate family is dependent, bilinearity gives a representation with fewer terms; independence in both coordinates rules out one with fewer than k terms.Source: §9.1 · pp. 25–26

9.2 Laplace Transform Preliminaries

  • For finite signed measures on R with Laplace transforms finite on an open interval, the transforms extend analytically, and agreement on a set accumulating at 0 implies equality of the measures.The argument uses analyticity and the identity theorem to obtain agreement near 0, where transform equality identifies the measure.Source: §9.2 · pp. 27–28
  • For measures in Mexp(R2), equality of one-dimensional pushforwards in countably infinitely many directions in S1 identifies the full finite signed measure.This extends the projection-identification result to signed measures; the Carleman condition supports the underlying uniqueness argument and is satisfied when the Laplace transform exists.Source: §9.2 · pp. 28–29
  • The Laplace transform of a factorizable measure factors into transforms of its one-dimensional factors, and this structure extends to finite mixtures in Mexp(R2).The mixture result provides a structural ingredient for the proof of Theorem 1.Source: §9.2 · pp. 31–32

9.3 A Rank Criterion for Laplace-Transform Matrices and Proofs of Theorem 1 and Corollary 2

  • Linear independence of k measures is equivalent to full rank of a structured Laplace-transform evaluation matrix near zero for nonzero a, b with |a| ≠ 1.The matrix evaluates their Laplace transforms at geometrically spaced arguments.Source: §9.3 · pp. 32–33
  • The proof establishes injectivity by determining an unknown Laplace-transform entry from the rank constraint, including when the measures have rank below k.For lower rank, it reduces to k* = max{ℓ, ℓ′} and applies the already-established theorem with that parameter.Source: §9.3 · pp. 35–36
  • For finite nonnegative measures, failure of Laplace-transform finiteness near zero is preserved under convolution with any nonzero positive measure.This property is used in the proof of Corollary 2.Source: §9.3 · p. 37
  • For directions with both coordinates nonzero, a product measure’s projected pushforward is the convolution of the scaled factor measures, and Lemma 27 supplies the contradiction used in the proof.The argument concludes that the relevant product is not in M+ exp(R).Source: §9.3 · pp. 37–38
  • After replacing any remaining zero terms by 0 × 0, Theorem 1 applies and yields µ = µ′.Source: §9.3 · p. 38

10 Proofs of Topological and Geometric Results

This section proves topological and geometric results for low-rank measure classes using compactness, continuity, and silhouette-operator injectivity. It also uses the rank-≤2k bound on differences of rank-≤k elements and finite-dimensionality of S to ensure the restricted operator norm is finite.

10.1 Topological Properties of Low-Rank Measure Classes

  • For compact K1,K2, rank-≤k measures are weak∗closed (hence weakly and norm closed), proved via tensor rank closedness under an approximation-property condition.For the measure result, C(K1)^* has the approximation property.Source: §10.1 · pp. 38, 40, 43, 44

10.2 Proofs of Theorem 3 and Proposition 4

  • Theorem 3 follows by restricting the silhouette operator to a compact bounded rank-≤k class, where it is continuous and injective, yielding a weak∗-weak∗ homeomorphism onto its image.The operator is norm-to-norm, weak-to-weak, and weak∗-to-weak∗ continuous.Source: §10.2 · pp. 44–45
  • For Proposition 4, the proof uses compactness and injectivity to rule out a normalized sequence converging to a nonzero rank-≤2k measure in the silhouette operator’s kernel.The contradiction establishes the existence of a positive κ in the proof’s subsequent bound.Source: §10.2 · p. 45
  • For μ, μ′ in S ∩ R_≤k(K1, K2), μ − μ′ lies in S ∩ R_≤2k(K1, K2), so M = ||V|S|| is finite because S is finite dimensional.Source: §10.2 · p. 45

11 Proofs for Higher-Dimensional Extensions of the Silhouette Framework

The higher-dimensional extension proves identifiability from pairwise marginals by induction over coordinates. Its proof uses connectedness to propagate marginal agreement and rank and component-matching arguments to recover the product components.

11 Proofs for Higher-Dimensional Extensions of the Silhouette Framework

  • Connectedness ensures every coordinate has an observed edge marginal, supplying the one-dimensional marginal agreement used to start the induction.The proof then reduces to a spanning tree and removes a leaf so the induction applies to the remaining coordinates.Source: §11 · p. 46
  • Under joint irreducibility and probability-measure assumptions, linear independence gives rank k, forcing the alternative component indexing to be a permutation.The proof uses rank preservation and linear independence of the rank-one terms to match components.Source: §11 · pp. 47–48
  • Singularity and nonnegativity rule out mixing the distinguished singular component with other components across the two coordinates.The proof derives nonnegative coefficients for the singular components and obtains a contradiction if a primed component mixes them.Source: §11 · pp. 49–50
  • Once a component is matched on the first two coordinates, equality of its pairwise marginals identifies its remaining coordinate factors.For each further coordinate, the proof uses an edge marginal linking it to one of the first two coordinates.Source: §11 · p. 51

12 Proofs of the Sharpness Results

The sharpness proofs construct measures with matching or vanishing projected marginals, including distinct probability models and smooth, rapidly decaying examples. They also use a sequence whose projections converge to zero while its norms diverge to show the induced map is not weak*-to-weak* continuous, with the argument extending to rank-≤1 signed measures with C∞ densities.

Proposition 37

  • For m prescribed directions, Proposition 37 constructs a nonzero compactly supported measure of rank at most m + 1 that is annihilated by those projections and is a difference of positive measures with equal nonzero mass.The positive measures have compactly supported nonnegative factors.Source: §12 · pp. 53, 54, 56

Theorem 8 proof

  • Applying Proposition 37 with m = 2k − 1 and normalizing its positive measures yields distinct probability measures for the sharpness construction.Smoothing preserves their distinction because the difference remains nonzero after convolution.Source: §12 · pp. 56–57

Proposition 9 proof

  • Symmetry of paired directions yields a nonzero measure annihilated by all 2k projections, expressible as a difference of mixtures with at most ⌊k/2⌋ + 1 components each.After smoothing, the models remain distinct and retain the prescribed projection equalities.Source: §12 · pp. 57–58

Lemma 10 and Corollary 11 proofs

  • For any finite collection of directions, the construction gives nonzero signed factors with Schwartz densities whose product is annihilated by every projection, yielding distinct rank-≤2 probability mixtures with matching marginals.The resulting mixture densities decay faster than any polynomial.Source: §12 · p. 60

12 Proofs of the Sharpness Results

  • V(x_n) converges to zero in norm while ||x_n|| = n, so the induced map is not weak*-to-weak* continuous and V is not a weak*-to-weak* homeomorphic embedding.The proof uses the implication that continuity at zero would contradict the diverging norms.Source: §12 · p. 62
  • The same argument applies when the domain is restricted to rank-≤1 signed measures admitting C∞ densities.Every constructed x_n admits a C∞ density.Source: §12 · p. 62

A Equivalence of RIP, RSC, and Injectivity

For linear maps on matrix spaces, injectivity on rank-≤k matrices is equivalent to restricted strong convexity on rank-≤2k matrices and, after rescaling, to restricted isometry there. The proof establishes the implications through low-rank differences and a uniform lower bound.

A Equivalence of RIP, RSC, and Injectivity

  • Injectivity on rank-≤k matrices, restricted strong convexity on rank-≤2k matrices, and rescaled restricted isometry on rank-≤2k matrices are equivalent.The appendix states the equivalence for linear maps between finite-dimensional normed spaces.Source: A Equivalence of RIP, RSC, and… · p. 63
  • Injectivity on rank-≤k matrices gives a nonzero image for every nonzero rank-≤2k matrix, yielding a uniform lower bound on that rank class.A rank-≤2k matrix can be written as a difference of two rank-≤k matrices; compactness then gives the lower bound.Source: A Equivalence of RIP, RSC, and… · p. 63
  • Restricted strong convexity on rank-≤2k matrices implies restricted isometry there after rescaling the operator.The proof uses the finite operator norm to obtain the rescaling.Source: A Equivalence of RIP, RSC, and… · p. 63
  • Restricted isometry on rank-≤2k matrices implies injectivity on rank-≤k matrices because differences of rank-≤k matrices have rank at most 2k.Source: A Equivalence of RIP, RSC, and… · p. 63

B.1 Implementation and Runtime Details

The experiments use dataset-specific train, validation, and test splits, with rounding and incremental updates to reduce SME computation. The appendix also specifies the MAF architectures and validation-based stopping procedure.

B.1 Implementation and Runtime Details

  • HEPMASS used 250 training and 250 validation samples with a fixed 20,000-sample test set; kin8nm used 1,000 training and 300 validation samples, with remaining samples tested per split.The splits balance computational cost and stability of model selection.Source: §B.1 · p. 64
  • Rounding coordinates only for SME partitioning enabled weighted empirical measures and reduced Wasserstein computation, while KDEs used the original samples.Rounding was to multiples of 0.125 for HEPMASS and 0.25 for kin8nm; preliminary tests found no noticeable density-estimation effect.Source: §B.1 · p. 64
  • With incremental updates and nine CPU cores, one HEPMASS candidate-label evaluation took about 6–7 ms, while one kin8nm evaluation took about 3–4 ms.HEPMASS objective evaluations average Wasserstein distances across 168 projection computations.Source: §B.1 · p. 64

B.1.1 MAF Implementation Details

  • MAF used five autoregressive transformations, dataset-specific hidden widths, and Adam training with validation-based early stopping.Widths were (64, 64) for HEPMASS and (32, 32) for kin8nm; training stopped after 10 or 30 non-improving validation epochs, respectively, and restored the best validation state.Source: §B.1.1 · p. 64

B.2 Additional Experimental Results

These additional HEPMASS experiments examine how the silhouette objective relates to downstream density-estimation performance during optimization. Test log-likelihood generally improves as the objective decreases, though a smaller-run average later plateaus and slightly declines.

B.2 Additional Experimental Results

  • The larger experiment required approximately 90 hours, with roughly half spent fitting mixture-of-Naive-Bayes KDE estimators and evaluating them.Each evaluation of all 79 alternative labels for one sample took approximately 4.5 seconds.Source: §B.2 · p. 65
  • On HEPMASS, test log-likelihood improves nearly monotonically as the silhouette objective falls during optimization.The experiment used 20,000 training samples and k=80 components; Figure 3a tracks the objective and test log-likelihood.Source: §B.2 · pp. 64–65, Figure 3 · p. 65
  • Across 20 HEPMASS runs with 500 training samples and k=5, log-likelihood improves with the falling objective, then plateaus and slightly declines late.The cause of the late decline is unclear.Source: §B.2 · p. 65

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