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arxiv: 2605.15688 · v1 · pith:MU5Q3RHHnew · submitted 2026-05-15 · 📊 stat.ML · cs.AI· cs.LG· math.PR

α-TCAV: A Unified Framework for Testing with Concept Activation Vectors

Ekkehard Schnoor , Jawher Said , Malik Tiomoko , Wojciech Samek , Alexander Jung This is my paper

Pith reviewed 2026-05-19 19:44 UTC · model grok-4.3

classification 📊 stat.ML cs.AIcs.LGmath.PR
keywords Concept Activation VectorsTCAVExplainable AIStatistical AnalysisSensitivity ScoresProbabilistic FormulationDeep LearningVariance Reduction
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The pith

The standard TCAV score has non-decaying variance from its discontinuous indicator, which α-TCAV fixes by substituting a parameterized smooth function for a unified probabilistic framework.

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

The paper derives distributions for several classes of Concept Activation Vectors and shows that the classic TCAV score relies on a discontinuous indicator function. This choice produces variance that fails to decay in the regimes where the method is most needed. α-TCAV replaces the indicator with a smooth tunable function, creating a single probabilistic formulation that contains both ordinary TCAV and Multi-TCAV as special cases. The framework supplies explicit guidance for choosing the parameter to match Multi-TCAV behavior at lower cost or to obtain a calibrated Bayes-optimal measure of concept influence. The analysis also yields a practical recommendation to devote the entire sampling budget to one CAV rather than dividing it across many.

Core claim

The central claim is that the discontinuous indicator inside the standard TCAV score induces non-decaying variance in critical regimes. Replacing that indicator with a parameterized smooth function produces α-TCAV, a unified probabilistic formulation that subsumes both TCAV and Multi-TCAV, admits closed-form distributions for the resulting sensitivity scores, and supplies principled tuning rules for the smoothing parameter.

What carries the argument

The parameterized smooth function that replaces the discontinuous indicator inside the TCAV sensitivity score.

If this is right

  • Established choices for TCAV variants lack theoretical justification once the distributions are derived.
  • Tuning the smoothing parameter lets users imitate Multi-TCAV at substantially lower computational cost.
  • Alternative tuning yields a calibrated Bayes-optimal probabilistic measure of a concept's influence.
  • Allocating the full sampling budget to a single CAV produces better results than splitting the budget across several CAVs.

Where Pith is reading between the lines

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

  • Adopting α-TCAV could reduce the number of unstable explanations encountered when auditing real deployed models.
  • The same smoothing idea may transfer to other gradient-based attribution methods that currently rely on hard thresholds.
  • Empirical checks on large vision models would test whether the predicted variance reduction appears at practical sample sizes.

Load-bearing premise

The smooth function can be tuned to imitate Multi-TCAV or to recover a calibrated Bayes-optimal measure without introducing additional bias.

What would settle it

Measure the empirical variance of the α-TCAV sensitivity score as the number of samples grows in the regime where standard TCAV variance remains constant; the claim is settled if the new variance decays while the old one does not.

Figures

Figures reproduced from arXiv: 2605.15688 by Alexander Jung, Ekkehard Schnoor, Jawher Said, Malik Tiomoko, Wojciech Samek.

Figure 1
Figure 1. Figure 1: TCAV, Multi-TCAV and α-TCAV (our proposed framework). For input x, the gradient z = ∇hl,k(fl(x)) of logit k at layer l is projected onto a (random) CAV wCAV ∈ R d (at layer l), yielding the sensitivity scores ⟨z, wCAV⟩. Next, different TCAV scores are computed, where TCAV (left) uses a hard indicator on a single CAV, potentially leading to large (or even non-vanishing) variance. Multi-TCAV (middle) average… view at source ↗
Figure 2
Figure 2. Figure 2: The indicator function 1{x>0} and approximations by the sigmoid function s, the scaled sigmoid function sα, as well as the pointwise limit s∞(x) := limα→∞ sα(x), the Heaviside function. formal definition - that differs from the originally used indicator function only in the origin; compare also [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Schematic illustration: 1-TCAV corresponds to a Bernoulli distribution, while Multi-TCAV is associated to a (normalized) Binomial distribution. α-TCAV follows a logit-normal distribution, that can either be bimodal (Bernoulli-like; left), or unimodal (Binomial-like; right). 9 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of the variance between a Bernoulli distribution and a scaled (average of [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Numerical plot of the variance ratio function [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Numerical simulation of mean and variance (columns) of the different TCAV approaches in [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Wall-clock time per TCAV estimate on ResNet-50/DTD (PatternCAV, 6 layer [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Comparison of TCAV methods on ResNet-50 (layer 2, PatternCAV; DTD textures, [PITH_FULL_IMAGE:figures/full_fig_p020_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Effect of total sample budget [PITH_FULL_IMAGE:figures/full_fig_p021_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Illustration of the one-dimensional Gaussian distributions of the classification score [PITH_FULL_IMAGE:figures/full_fig_p022_10.png] view at source ↗
Figure 10
Figure 10. Figure 10: Notably, the classification accuracy is determined solely by the scalars [PITH_FULL_IMAGE:figures/full_fig_p024_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Comparison of theoretical prediction and histogram of empirical simulation of the [PITH_FULL_IMAGE:figures/full_fig_p028_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Classification score distributions of PatternCAV on ResNet-50 (rows: the concepts striped, zigzagged and dotted; columns: layer2 to layer4). Histograms show the empirical density of g(x) = w⊤ CAVx, where x denotes the laten activation of either non-concept (C1, red, 1000 samples) or concept (C2, blue, 50 samples) classes; solid curves are fitted Gaussians (theoretical prediction in advance — like in [PIT… view at source ↗
Figure 4
Figure 4. Figure 4: Therefore, this is is trivially also the generic upper bound (21) for the variance of any TCAV [PITH_FULL_IMAGE:figures/full_fig_p036_4.png] view at source ↗
read the original abstract

Concept Activation Vectors (CAVs) are a fundamental tool for concept-based explainability in deep learning, yet their practical utility is limited by statistical instability. We analyze the stochastic nature of CAVs and the Testing with CAVs (TCAV) method, deriving the distributions of major CAV classes including PatternCAV, FastCAV, and ridge regression-based CAVs. We then identify a fundamental flaw in the standard TCAV score: its reliance on a discontinuous indicator function induces non-decaying variance in critical regimes. To address this, we introduce $\alpha$-TCAV, a generalized framework that replaces the indicator with a parameterized smooth function, yielding a unified probabilistic formulation that subsumes both TCAV and Multi-TCAV. We characterize the induced distributions of sensitivity scores and different TCAV variants, showing that established state-of-the-art choices lack theoretical justification. We provide principled guidance on tuning the parameter in $\alpha$-TCAV -- either to imitate Multi-TCAV at substantially lower computational cost, or to obtain a calibrated Bayes-optimal probabilistic measure of a concept's influence. Finally, our analysis yields practical recommendations that challenge established routines: most notably, allocating the full sampling budget to a single CAV rather than splitting it across several.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript analyzes the stochastic properties of Concept Activation Vectors (CAVs) including PatternCAV, FastCAV, and ridge-regression variants, derives their distributions, and identifies a fundamental flaw in standard TCAV: the discontinuous indicator function produces non-decaying variance near decision boundaries or in low-signal regimes. It introduces α-TCAV, which replaces the indicator with a parameterized smooth function to obtain a unified probabilistic formulation subsuming TCAV and Multi-TCAV, characterizes the resulting sensitivity-score distributions, supplies tuning guidance for the α parameter (to imitate Multi-TCAV at lower cost or to achieve a calibrated Bayes-optimal measure), and recommends allocating the full sampling budget to a single CAV rather than splitting it.

Significance. If the analytic derivations of the CAV distributions and the claim that the smooth-function replacement introduces no new finite-sample bias both hold, the work supplies the first rigorous probabilistic account of TCAV instability and a principled way to stabilize it. The explicit variance-flaw diagnosis and the practical recommendation to use a single CAV constitute concrete, falsifiable advances that could improve reliability of concept-based explanations; the unified framework also offers a route to lower-cost Multi-TCAV emulation.

major comments (2)
  1. [stochastic analysis section] Stochastic analysis section: the central claim that the distributions for PatternCAV, FastCAV, and ridge-regression CAVs are analytically tractable (and therefore permit exact tuning guidance for α) rests on regularity conditions that are not fully stated; any hidden approximation or unverified moment condition would propagate directly into the assertion that established routines lack justification and into the Bayes-optimal calibration claim.
  2. [α-TCAV framework section] α-TCAV framework section: the replacement of the discontinuous indicator by the parameterized smooth function is asserted to preserve the exact sensitivity distribution without introducing new bias, yet the specific functional form chosen for the smooth surrogate may implicitly encode earlier empirical choices; explicit error bounds or a finite-sample bias analysis is required to support the claim that α can be tuned to a calibrated posterior without distortion.
minor comments (2)
  1. The abstract states that α-TCAV yields “substantially lower computational cost” than Multi-TCAV, but no quantitative runtime or sample-complexity comparison appears in the main text or experiments.
  2. Notation for the sensitivity score under different CAV estimators is introduced without a consolidated table; a single reference table would improve readability when comparing the derived distributions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review. The comments correctly identify areas where greater explicitness is needed to support the central claims. We respond to each major comment below and commit to revisions that strengthen the rigor of the stochastic analysis and the α-TCAV framework without altering the manuscript's core contributions.

read point-by-point responses

  1. Referee: [stochastic analysis section] Stochastic analysis section: the central claim that the distributions for PatternCAV, FastCAV, and ridge-regression CAVs are analytically tractable (and therefore permit exact tuning guidance for α) rests on regularity conditions that are not fully stated; any hidden approximation or unverified moment condition would propagate directly into the assertion that established routines lack justification and into the Bayes-optimal calibration claim.

    Authors: We agree that the regularity conditions underlying the closed-form distributions must be stated explicitly. The derivations in the stochastic analysis section rely on standard assumptions for linear and ridge estimators (finite second moments of activations, full-rank covariance in the relevant subspace, and Gaussian or sub-Gaussian tails for concentration), but these were not collected in one place. In the revision we will insert a dedicated paragraph at the start of the section that enumerates all required conditions, including moment bounds and regularity requirements for the PatternCAV, FastCAV, and ridge-regression estimators. With these conditions visible, the claims about lack of justification for prior routines and the validity of Bayes-optimal tuning guidance will rest on a transparent foundation. revision: yes

  2. Referee: [α-TCAV framework section] α-TCAV framework section: the replacement of the discontinuous indicator by the parameterized smooth function is asserted to preserve the exact sensitivity distribution without introducing new bias, yet the specific functional form chosen for the smooth surrogate may implicitly encode earlier empirical choices; explicit error bounds or a finite-sample bias analysis is required to support the claim that α can be tuned to a calibrated posterior without distortion.

    Authors: The smooth surrogate is introduced as a continuous relaxation whose expectation recovers the original TCAV score in the limit as α → ∞, and the paper characterizes the resulting sensitivity-score distribution exactly under the same probabilistic model used for the CAV estimators. We do not claim the finite-α version is bias-free for every possible surrogate; the functional form is chosen for analytic tractability and monotonicity. To meet the referee's request we will add a short finite-sample bias analysis and approximation-error bounds in the α-TCAV framework section, showing that the bias term is O(1/α) under the stated moment conditions and vanishes uniformly away from the decision boundary. This will also clarify that the calibration of α to a Bayes-optimal posterior remains valid once the controlled approximation error is accounted for. revision: yes

Circularity Check

0 steps flagged

Derivations of CAV distributions and α-TCAV replacement are self-contained first-principles analysis.

full rationale

The paper derives distributions for PatternCAV, FastCAV, and ridge-regression CAVs directly from stochastic properties of the underlying models and then replaces the discontinuous indicator in TCAV with a parameterized smooth function to obtain α-TCAV. No step reduces a claimed prediction or uniqueness result to a fitted parameter or prior self-citation by construction. The analytic tractability statements and tuning guidance follow from the stated regularity conditions on the sensitivity scores rather than from re-labeling inputs as outputs. The central claim about non-decaying variance is therefore an independent consequence of the indicator discontinuity and is not forced by the paper's own definitions or citations.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claims rest on analytic tractability of CAV distributions under standard assumptions and on the existence of a tunable smooth function that preserves interpretability while reducing variance.

free parameters (1)
  • alpha
    Smoothing parameter that controls the transition from hard indicator to probabilistic score; its value determines whether the method imitates Multi-TCAV or targets Bayes optimality.
axioms (1)
  • domain assumption CAVs of PatternCAV, FastCAV, and ridge-regression types possess analytically derivable distributions under the stochastic model of activations.
    Invoked when the paper states it derives the distributions of major CAV classes.
invented entities (1)
  • α-TCAV sensitivity score no independent evidence
    purpose: Probabilistic measure of concept influence obtained via the smooth replacement function.
    New quantity introduced to replace the discontinuous TCAV score.

pith-pipeline@v0.9.0 · 5767 in / 1449 out tokens · 38769 ms · 2026-05-19T19:44:12.106838+00:00 · methodology

discussion (0)

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