Learning Arithmetic Formulas in the Presence of Noise: A General Framework and Applications to Unsupervised Learning

Abstract: We present a general framework for designing efficient algorithms for unsupervised learning problems, such as mixtures of Gaussians and subspace clustering. Our framework is based on a meta algorithm that learns arithmetic circuits in the presence of noise, using lower bounds. This builds upon the recent work of Garg, Kayal and Saha (FOCS 20), who designed such a framework for learning arithmetic circuits without any noise. A key ingredient of our meta algorithm is an efficient algorithm for a novel problem called Robust Vector Space Decomposition. We show that our meta algorithm works well when certain matrices have sufficiently large smallest non-zero singular values. We conjecture that this condition holds for smoothed instances of our problems, and thus our framework would yield efficient algorithms for these problems in the smoothed setting.

• The paper introduces a unified framework that reduces unsupervised learning to robust reconstruction of structured arithmetic formulas in noisy settings.
• It employs noise-robust vector space decomposition and linear operators to efficiently separate and recover model components.
• Empirical results and theoretical bounds indicate improved parameter recovery in applications like mixture models and subspace clustering.

Learning Arithmetic Formulas in the Presence of Noise: General Framework and Unsupervised Learning Applications

Introduction and Problem Motivation

This work develops a unified framework for learning arithmetic formulas in noisy environments, focusing on applications in unsupervised learning—specifically, mixture models (such as mixtures of Gaussians) and subspace clustering. The methodology extends the algebraic meta-algorithm of [Garg, Kayal, Saha, FOCS '20] to address the fundamental challenge of noise-tolerance, which prior work left unsolved. The general principle is that for a broad class of unsupervised problems where data has algebraic structure, the task of model parameter recovery reduces to robust reconstruction of sums of structured polynomials (arithmetic formulas) given only approximate empirical moments.

Algebraic Reduction and the Role of Noise

Many unsupervised problems (e.g., mixture models, ICA, Hidden Markov Models, topic models) can be reformulated as the problem of reconstructing arithmetic formulas from low-order moments. For mixtures of (zero-mean) Gaussians, the relevant polynomial is $$f(x) = \mathbb{E}_{a \sim \mathcal{D}}[\langle x, a\rangle^d]$$, and the task is to decompose this as a sum of powers of quadratics. In real problems, only empirical moments are available, so $f(x)$ is observed up to a noise term $\eta(x)$ reflecting sample error and outliers.

Mathematically, the main object becomes: $f(x) = T_1(x) + \cdots + T_s(x) + \eta(x)$ where each $T_i$ is a "structured" polynomial (e.g., powers of linear or quadratic forms). The objective is to efficiently and robustly recover the $T_i$, even in the presence of noise.

The Meta-Algorithm: Lower Bounds and Decomposition

The framework derives from an intersection of algebraic complexity and numerical analysis:

• Key Ingredients:
  • Construction of linear operators $L$ and $B$ (borrowed from arithmetic circuit lower bounds) that "separate" the summands $T_i$.
  • Application of $L$ and $B$ yields a direct-sum decomposition of polynomial coefficient spaces, provided certain non-degeneracy and singular value conditions hold.
  • The main technical step is a robust Vector Space Decomposition (VSD) problem: given $B$ acting on $U = U_1 \oplus ... \oplus U_s$, decomposing $U$ efficiently when the data is noisy.
• Noise-Robustness Analysis:
  • The algorithmic guarantees depend polynomially on the inverse of the smallest nonzero singular values of certain "moment" and "adjoint algebra" operators.
  • The central correctness theorem asserts that if the smallest relevant singular values and the "separation" of component subspaces (in the sense of condition numbers) are bounded below, the method yields recovery error in the output $T_i$ that is at most polynomially larger than the input noise magnitude (Theorem~\ref{thm:robustCircuitReconstruction}).
  • The framework is shown (conjecturally) to be effective in smoothed analysis regimes where noise and random perturbations ensure generic invertibility.

Robust Vector Space Decomposition and Adjoint Algebras

A core technical contribution is the robust algorithm for VSD, generalized to what the authors term Robust Vector Space Decomposition (RVSD). The approach:

• Employs the theory of adjoint algebras (a generalization of centralizer algebras) to identify decompositions via the eigenstructure of associated linear operators.
• Introduces a noise-robust variant by analyzing perturbations in the action of $B$ on $U$, leveraging tools such as Weyl's inequality and Wedin's theorem on singular value perturbations.
• Achieves polynomial-time implementation via SVD and pseudo-inverse computations rather than explicit enumeration or combinatorial search.
• Tolerates random or adversarial noise levels up to basic numerical limits determined by the minimal singular values of problem instance matrices.

Applications

Subspace Clustering

• Reduces clustering of points lying near a union of subspaces to a decomposition problem on symmetric tensor powers of the data points.
• Under suitable direct sum and indecomposability conditions, RVSD efficiently recovers the underlying subspaces.
• Theoretical guarantees are given in terms of polynomial bounds on the recovery error as a function of coherence, condition numbers, and smallest singular values of moment matrices constructed from the data (Theorem~\ref{thm:sc}).
• Smoothed analysis gives high-probability bounds in random (smoothed) models, relating stability to canonical angles between subspaces.

Mixture of Gaussians

• Extends prior tensor decomposition-based techniques, providing a direct algebraic reduction for parameter learning in mixtures of (zero-mean) Gaussians.
• The method avoids some previous algorithmic steps (such as multi-GCD and explicit tensor desymmetrization) in favor of direct robust vector space decomposition of suitable functionals of the data.
• The analysis covers a broad parameter regime based on singular value lower bounds, and conjectures are made for further extending guarantees in the general mean/covariance case.

Strong Claims and Empirical Indications

• The error bound for adversarial noise is shown to scale as $$\operatorname{poly}(s, d, \kappa, 1/\sigma, 1/\delta) \cdot \epsilon$$, where $\kappa$ is a suitable condition number and $\sigma$ is the minimal relevant singular value.
• Empirical results (not detailed, but referenced) suggest that for tensor decomposition the proposed robust adjoint algebra methods can outperform existing techniques in terms of noise-tolerance.
• The framework rigorously explains, and in some parameter regimes unifies, a diverse collection of unsupervised learning algorithms as instances of algebraic decomposition tasks.

Theoretical and Practical Implications

This work provides a principled, algebraically-motivated path toward robust and broadly applicable unsupervised learning algorithms, pushing beyond worst-case intractability toward guarantees in smoothed/random settings. The explicit connection between singular value analysis, arithmetic circuit lower bounds, and component separation in noisy data provides both a unifying perspective and guidance for the construction of future algorithms.

The framework supports extensions not only to classical problems (mixtures, clustering) but also points toward robust approaches for topic modeling, learning mixtures of polynomial transformations, and potentially for algorithms tolerant to severe outliers—with these directions suggested as open problems. The theory also links to operator scaling and invariant theory (e.g., via suggestions to use operator scaling to construct canonical inner products for orthogonality).

Future Directions

• Determination of minimal singular value lower bounds in concrete learning scenarios, particularly in mixed or heavily perturbed cases.
• Extension and smoothed analysis for general mixture models (with arbitrary means/covariances), especially for the algorithm's performance under random (smoothed) instance models.
• Algorithmic refinements to reduce computational complexity through direct exploitation of additional algebraic structure, e.g., via operator scaling or carefully designed projections.
• Integration and empirical benchmarking of the framework for practical datasets and high-dimensional instances.

Conclusion

The authors develop a robust, algebraically justified algorithmic framework for unsupervised learning via arithmetic formula reconstruction in the presence of noise, with proven and conjectured guarantees for many central problems in the field. The results formally connect algebraic complexity, numerical stability, and unsupervised model estimation, and the analysis yields both new algorithmic tools and conceptual clarity regarding when efficient, noise-tolerant learning is possible.

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Reference: "Learning Arithmetic Formulas in the Presence of Noise: A General Framework and Applications to Unsupervised Learning" (2311.07284)