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component-wise bounds

Component-wise bounds are mathematical limits established individually for each entry, element, or coordinate of a vector, matrix, or multi-dimensional array, rather than for the entire structure as a single aggregate quantity. Unlike norm-wise bounds that summarize overall magnitude across all dimensions using a single scalar value, component-wise bounds provide localized constraints on the minimum, maximum, or absolute values that specific components can assume. This granular specification is widely used in numerical linear algebra, interval arithmetic, error analysis, and algorithm convergence studies to track dimensional variation, control perturbation propagation, and guarantee performance on an element-by-element basis.

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Value bounds and Convergence Analysis for Averages of LRP attributions

Value bounds and Convergence Analysis for Averages of LRP attributions

Alexander Binder, Nastaran Takmil-Homayouni, Urun Dogan

OrganizationsLeipzig UniversityMicrosoftOtto-von-Guericke-Universität Magdeburg

Why you should read this

Establishes formal value bounds and convergence guarantees for averaged Layer-wise Relevance Propagation attributions, demonstrating that LRP-beta uniquely avoids sensitivity to weight norms in smoothed and data-augmented model explanations.

We analyze numerical properties of Layer-wise relevance propagation (LRP)-type attribution methods by representing them as a product of modified gradient matrices. This representation creates an analogy to matrix multiplications of Jacobi-matrices which arise from the chain rule of differentiation. In order to shed light on the distribution of attribution values, we derive upper bounds for singular values. Furthermore we derive component-wise bounds for attribution map values. As a main result, we apply these component-wise bounds to obtain multiplicative constants. These constants govern the convergence of empirical means of attributions to expectations of attribution maps. This finding has important implications for scenarios where multiple non-geometric data augmentations are applied to individual test samples, as well as for Smoothgrad-type attribution methods. In particular, our analysis reveals that the constants for LRP-beta remain independent of weight norms, a significant distinction from both gradient-based methods and LRP-epsilon.

Added

2026-09-29