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RACH-Space: Reconstructing Adaptive Convex Hull Space with Applications in Weak Supervision

Woojoo Na, Abiy Tasissa

TL;DR

RACH-Space tackles weakly supervised learning where only incomplete, noisy weak signals are available. It introduces a geometry-based framework that treats weak signals as columns of a matrix and leverages nested convex hulls to define a safe region guiding label estimation, parameterized by a single $\epsilon$ that bounds per-signal error rates. The method formulates a quadratic program to recover a probabilistic label vector under a unit-sum constraint, and uses convex-hull structure to regularize the solution, with efficiency enhanced via signal grouping. Empirically, RACH-Space achieves state-of-the-art or on-par performance on 14 real-world WRENCH datasets, demonstrating robustness without strong distributional assumptions.

Abstract

We introduce RACH-Space, an algorithm for labelling unlabelled data in weakly supervised learning, given incomplete, noisy information about the labels. RACH-Space offers simplicity in implementation without requiring hard assumptions on data or the sources of weak supervision, and is well suited for practical applications where fully labelled data is not available. Our method is built upon a geometrical interpretation of the space spanned by the set of weak signals. We also analyze the theoretical properties underlying the relationship between the convex hulls in this space and the accuracy of our output labels, bridging geometry with machine learning. Empirical results demonstrate that RACH-Space works well in practice and compares favorably to the best existing label models for weakly supervised learning.

RACH-Space: Reconstructing Adaptive Convex Hull Space with Applications in Weak Supervision

TL;DR

RACH-Space tackles weakly supervised learning where only incomplete, noisy weak signals are available. It introduces a geometry-based framework that treats weak signals as columns of a matrix and leverages nested convex hulls to define a safe region guiding label estimation, parameterized by a single that bounds per-signal error rates. The method formulates a quadratic program to recover a probabilistic label vector under a unit-sum constraint, and uses convex-hull structure to regularize the solution, with efficiency enhanced via signal grouping. Empirically, RACH-Space achieves state-of-the-art or on-par performance on 14 real-world WRENCH datasets, demonstrating robustness without strong distributional assumptions.

Abstract

We introduce RACH-Space, an algorithm for labelling unlabelled data in weakly supervised learning, given incomplete, noisy information about the labels. RACH-Space offers simplicity in implementation without requiring hard assumptions on data or the sources of weak supervision, and is well suited for practical applications where fully labelled data is not available. Our method is built upon a geometrical interpretation of the space spanned by the set of weak signals. We also analyze the theoretical properties underlying the relationship between the convex hulls in this space and the accuracy of our output labels, bridging geometry with machine learning. Empirical results demonstrate that RACH-Space works well in practice and compares favorably to the best existing label models for weakly supervised learning.
Paper Structure (18 sections, 2 theorems, 4 equations, 1 figure, 4 tables, 1 algorithm)

This paper contains 18 sections, 2 theorems, 4 equations, 1 figure, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Overview of RACH-Space
  3. Contributions
  4. Related Work
  5. RACH-Space
  6. Problem Formulation
  7. Upper Bound on eps
  8. Using Convex Hull Structure to Update eps
  9. Algorithm
  10. Time Complexity & Adaptations
  11. Safe Region
  12. Solution Space inside conv(H1)
  13. Restricting the Solution Space inside conv(H1)
  14. Experiments
  15. Experiment 1: Comparison to Other Baselines
  16. ...and 3 more sections

Key Result

Theorem 5.1

For $\frac{\widetilde{\mathbf{b}}}{n} \in \textrm{Conv}(\textrm{Col}(\mathbf{A}))$ and $nk>m+1$, the optimisation program in eq:least_squares_b has infinitely many solutions.

Figures (1)

  • Figure 1: An illustration for the main idea of RACH-Space. Using the upper bound $\frac{2}{k}-\frac{2}{k^2}$ for $\bm{\epsilon}$, a right hand side vector $\widetilde{\mathbf{b}}$ is initialized. Our aim is to update the parameter $\bm{\epsilon}$ so that $\frac{\widetilde{\mathbf{b}}}{n}$ lies in the safe region, which is the region inside the convex hull but outside the second largest convex hull.

Theorems & Definitions (9)

  • Remark 3.1
  • Remark 3.2
  • Definition 3.3
  • Remark 3.4
  • Theorem 5.1
  • Definition 5.2
  • proof
  • Lemma 5.3
  • proof