# Source-Side Sufficiency for the Information Bottleneck: Exact Reduction and Finite-Block Equivalence

arXiv: https://arxiv.org/abs/2604.26744 · HTML: https://mlsanity.com/paper/2604.26744/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 35 references that are in the map.
- First posted on arXiv 29 Apr 2026.

## Builds on (most cited first, 8 of 8)

- [Deep Variational Information Bottleneck](https://mlsanity.com/paper/1612.00410/) (2016)
- [Emergence of Invariance and Disentanglement in Deep Representations](https://mlsanity.com/paper/1706.01350/) (2017)
- [Caveats for information bottleneck in deterministic scenarios](https://mlsanity.com/paper/1808.07593/) (2018)
- [On the Information Bottleneck Problems: Models, Connections, Applications and Information Theoretic Views](https://mlsanity.com/paper/2002.00008/) (2020)
- [Bottleneck Problems: Information and Estimation-Theoretic View](https://mlsanity.com/paper/2011.06208/) (2020)
- [Gaussian Lower Bound for the Information Bottleneck Limit](https://mlsanity.com/paper/1711.02421/) (2017)
- [Generalizing Bottleneck Problems](https://mlsanity.com/paper/1802.05861/) (2018)
- [Partial information decomposition: redundancy as information bottleneck](https://mlsanity.com/paper/2405.07665/) (2024)

## Built on by (most cited first, 1 of 1)

- [On Task Scope and Information Retention in Source Coding](https://mlsanity.com/paper/2609.37575/) (2026)

## Abstract (arXiv)

The input side of the Information Bottleneck may contain task-irrelevant variation that still costs rate. We identify this cost exactly. Let T be a source and C a relevance variable. Suppose the deterministic statistic Z = phi(T) satisfies C-Z-T. For any encoder p(X|T), its conditional average over the fibres of phi preserves I(X;C) and lowers the rate by I(X;T|Z). The reverse pullback preserves both coordinates. These maps establish equality of the relevance-rate curves and Lagrangian infima on standard Borel spaces for every tradeoff parameter. They also characterise all attained optima. Every full-source minimiser factors through Z, and every reduced minimiser pulls back to T. When C is finite and distortion is logarithmic loss, replacement of T^n by Z^n also preserves the optimal remote distortion at every blocklength and message budget. The operational rate-distortion functions are therefore equal.