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Mean Estimation from Coarse Data: Characterizations and Efficient Algorithms

Alkis Kalavasis, Anay Mehrotra, Manolis Zampetakis, Felix Zhou, Ziyu Zhu

TL;DR

Gaussian mean estimation from coarse data is studied, where each true sample is drawn from a $d$-dimensional Gaussian distribution with identity covariance, but is revealed only through the set of a partition containing $x$.

Abstract

Coarse data arise when learners observe only partial information about samples; namely, a set containing the sample rather than its exact value. This occurs naturally through measurement rounding, sensor limitations, and lag in economic systems. We study Gaussian mean estimation from coarse data, where each true sample $x$ is drawn from a $d$-dimensional Gaussian distribution with identity covariance, but is revealed only through the set of a partition containing $x$. When the coarse samples, roughly speaking, have ``low'' information, the mean cannot be uniquely recovered from observed samples (i.e., the problem is not identifiable). Recent work by Fotakis, Kalavasis, Kontonis, and Tzamos [FKKT21] established that sample-efficient mean estimation is possible when the unknown mean is identifiable and the partition consists of only convex sets. Moreover, they showed that without convexity, mean estimation becomes NP-hard. However, two fundamental questions remained open: (1) When is the mean identifiable under convex partitions? (2) Is computationally efficient estimation possible under identifiability and convex partitions? This work resolves both questions. [...]

Mean Estimation from Coarse Data: Characterizations and Efficient Algorithms

TL;DR

Gaussian mean estimation from coarse data is studied, where each true sample is drawn from a -dimensional Gaussian distribution with identity covariance, but is revealed only through the set of a partition containing .

Abstract

Coarse data arise when learners observe only partial information about samples; namely, a set containing the sample rather than its exact value. This occurs naturally through measurement rounding, sensor limitations, and lag in economic systems. We study Gaussian mean estimation from coarse data, where each true sample is drawn from a -dimensional Gaussian distribution with identity covariance, but is revealed only through the set of a partition containing . When the coarse samples, roughly speaking, have ``low'' information, the mean cannot be uniquely recovered from observed samples (i.e., the problem is not identifiable). Recent work by Fotakis, Kalavasis, Kontonis, and Tzamos [FKKT21] established that sample-efficient mean estimation is possible when the unknown mean is identifiable and the partition consists of only convex sets. Moreover, they showed that without convexity, mean estimation becomes NP-hard. However, two fundamental questions remained open: (1) When is the mean identifiable under convex partitions? (2) Is computationally efficient estimation possible under identifiability and convex partitions? This work resolves both questions. [...]
Paper Structure (54 sections, 30 theorems, 100 equations, 2 figures, 1 algorithm)

This paper contains 54 sections, 30 theorems, 100 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Setup.
  3. The Need for Convex Partitions.
  4. Prior Sample Complexity Results.
  5. Our Contributions.
  6. Characterization Techniques.
  7. Algorithmic Techniques.
  8. Related Works
  9. Learning from Coarse Data.
  10. Sheppard's Corrections.
  11. Learning from Partial Labels.
  12. Preliminaries
  13. Notation.
  14. The Coarse Observations Model
  15. Our Results
  16. ...and 39 more sections

Key Result

Theorem 3.1

Fix any ${\mu^\star}\in \mathbb{R}^d$. A convex partition $\mathdutchcal{P}$ of $\mathbb{R}^d$ is not information preserving (i.e., not identifiable; def:identifiability) with respect to ${\mu^\star}$ if and only if there is a unit vector $v\in \mathbb{R}^d$ such that almost everyLet $\mathdutchcal{

Figures (2)

  • Figure 1: Each row shows a distribution (rows 1 and 2: Beta; rows 3 and 4: Gaussian). Columns (1,3): bar plots of empirical variances (red = original, blue = truncated) with variance ratio $r=\mathrm{Var}_{\text{trunc}}/\mathrm{Var}_{\text{orig}}$ annotated. Columns (2,4): histogram overlays (original vs. truncated) for half-line (col. 2) and interval $[L,U]$ truncations (col. 4). In all displayed runs, $r<1$, indicating variance reduction under these convex truncations.
  • Figure 2: Same layout as \ref{['fig:4']}. Rows 1 and 2: Laplace; rows 3 and 4: Quartic (density $\propto e^{-(x-\mu)^4/s}$). Columns (1,3) report variance bars and $r$; columns (2,4) show the corresponding histogram overlays for half-line and interval $[L,U]$ truncations. The observed $r<1$ across settings provides further evidence of practical variance reduction beyond the Gaussian case.

Theorems & Definitions (58)

  • Definition 1: Coarse Mean Estimation fotakis2021coarse
  • Definition 2: Information Preservation (a.k.a. Identifiability)
  • Definition 3: $\alpha$-Information Preservation; fotakis2021coarse
  • Definition 4: Slab
  • Theorem 3.1: Characterization of identifiability for convex partitions
  • Theorem 3.2: Coarse Mean Estimation Algorithm
  • Theorem 3.3: Informal; See \ref{['cor:friction:linearEstimation']}
  • Proposition 4.1: Likelihood-Based Characterization of Identifiability
  • proof : Proof of \ref{['thm:likelihood']}
  • Proposition 4.2: Convexity; fotakis2021coarse
  • ...and 48 more