Solving Random Quadratic Systems of Equations Is Nearly as Easy as Solving Linear Systems (1505.05114v2)

Abstract: We consider the fundamental problem of solving quadratic systems of equations in $n$ variables, where $y_i = |\langle \boldsymbol{a}_i, \boldsymbol{x} \rangle|^2$, $i = 1, \ldots, m$ and $\boldsymbol{x} \in \mathbb{R}^n$ is unknown. We propose a novel method, which starting with an initial guess computed by means of a spectral method, proceeds by minimizing a nonconvex functional as in the Wirtinger flow approach. There are several key distinguishing features, most notably, a distinct objective functional and novel update rules, which operate in an adaptive fashion and drop terms bearing too much influence on the search direction. These careful selection rules provide a tighter initial guess, better descent directions, and thus enhanced practical performance. On the theoretical side, we prove that for certain unstructured models of quadratic systems, our algorithms return the correct solution in linear time, i.e. in time proportional to reading the data ${\boldsymbol{a}_i}$ and ${y_i}$ as soon as the ratio $m/n$ between the number of equations and unknowns exceeds a fixed numerical constant. We extend the theory to deal with noisy systems in which we only have $y_i \approx |\langle \boldsymbol{a}_i, \boldsymbol{x} \rangle|^2$ and prove that our algorithms achieve a statistical accuracy, which is nearly un-improvable. We complement our theoretical study with numerical examples showing that solving random quadratic systems is both computationally and statistically not much harder than solving linear systems of the same size---hence the title of this paper. For instance, we demonstrate empirically that the computational cost of our algorithm is about four times that of solving a least-squares problem of the same size.

• The paper presents a novel two-step method that uses spectral initialization followed by adaptive nonconvex optimization to accurately solve quadratic equations.
• The paper demonstrates that its approach achieves provable near-linear time complexity, with computational costs roughly four times that of solving a least-squares problem.
• The paper extends its framework to noisy systems, achieving near-optimal accuracy and highlighting potential applications in phase retrieval and other nonconvex optimization challenges.

Solving Random Quadratic Systems of Equations Is Nearly as Easy as Solving Linear Systems

This paper addresses the problem of solving quadratic systems of equations, represented as $$y_i = |\langle \boldsymbol{a}_i, \boldsymbol{x} \rangle|^2$$, where $\boldsymbol{x} \in \mathbb{R}^n$ is unknown. The authors propose a novel method that begins with a spectral method to compute an initial guess and then minimizes a nonconvex functional using the adaptive technique inspired by the Wirtinger flow.

Key Features of the Proposed Method

The method introduces distinct objective functionals and update rules that adaptively drop terms excessively influencing the search direction. This careful selection ensures a tighter initial guess and more stable descent directions, enhancing practical performance.

The theoretical contributions of this paper include proving that for unstructured models of quadratic systems, the proposed algorithms provide correct solutions in linear time, proportional to reading the data, as long as the ratio $m/n$ (equations to unknowns) exceeds a fixed constant.

Moreover, the method extends to noisy systems, where only approximate measurements of the form $$y_i \approx |\langle \boldsymbol{a}_i, \boldsymbol{x} \rangle|^2$$ are available. Here, the algorithm achieves near-optimal statistical accuracy.

Practical and Theoretical Implications

The empirical results suggest that solving random quadratic systems is computationally and statistically not much harder than solving linear systems, often with the computational cost about four times that of solving a least-squares problem of the same size. The method's stability in the presence of noise confirms its robustness in practical scenarios.

The implications of this research are profound for fields requiring the extraction of phase information from magnitude-only observations, such as phase retrieval problems seen in X-ray crystallography and diffraction imaging.

Speculations on Future Developments

Looking forward, the approach could inspire similar techniques in tackling other nonconvex optimization problems. Particularly, its success underscores the potential of non-lifting procedures for high-dimensional data, possibly spurring advancements in areas like low-rank matrix recovery or tensor decomposition.

The exploration of other objective functions within this framework could yield alternative algorithms with applicability across numerous domains in signal processing and machine learning. Furthermore, adapting this approach to handle more complex noise models or structured designs could broaden its usability.

Conclusion

This paper significantly contributes to the field of nonconvex optimization by presenting a method that efficiently tackles quadratic systems of equations with both theoretical soundness and practical robustness. Future work may build on these insights to address broader classes of problems in information theory and data science.