On Learning Parallel Pancakes with Mostly Uniform Weights

Ilias Diakonikolas, Daniel Kane, Sushrut Karmalkar, Jasper C.H. Lee, Thanasis Pittas
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:13601-13621, 2025.

Abstract

We study the complexity of learning $k$-mixtures of Gaussians ($k$-GMMs) on $\mathbb R^d$. This task is known to have complexity $d^{\Omega(k)}$ in full generality. To circumvent this exponential lower bound on the number of components, research has focused on learning families of GMMs satisfying additional structural properties. A natural assumption posits that the component weights are not exponentially small and that the components have the same unknown covariance. Recent work gave a $d^{O(\log(1/w_{\min}))}$-time algorithm for this class of GMMs, where $w_{\min}$ is the minimum weight. Our first main result is a Statistical Query (SQ) lower bound showing that this quasi-polynomial upper bound is essentially best possible, even for the special case of uniform weights. Specifically, we show that it is SQ-hard to distinguish between such a mixture and the standard Gaussian. We further explore how the distribution of weights affects the complexity of this task. Our second main result is a quasi-polynomial upper bound for the aforementioned testing task when most of the weights are uniform while a small fraction of the weights are potentially arbitrary.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-diakonikolas25b, title = {On Learning Parallel Pancakes with Mostly Uniform Weights}, author = {Diakonikolas, Ilias and Kane, Daniel and Karmalkar, Sushrut and Lee, Jasper C.H. and Pittas, Thanasis}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {13601--13621}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/diakonikolas25b/diakonikolas25b.pdf}, url = {https://proceedings.mlr.press/v267/diakonikolas25b.html}, abstract = {We study the complexity of learning $k$-mixtures of Gaussians ($k$-GMMs) on $\mathbb R^d$. This task is known to have complexity $d^{\Omega(k)}$ in full generality. To circumvent this exponential lower bound on the number of components, research has focused on learning families of GMMs satisfying additional structural properties. A natural assumption posits that the component weights are not exponentially small and that the components have the same unknown covariance. Recent work gave a $d^{O(\log(1/w_{\min}))}$-time algorithm for this class of GMMs, where $w_{\min}$ is the minimum weight. Our first main result is a Statistical Query (SQ) lower bound showing that this quasi-polynomial upper bound is essentially best possible, even for the special case of uniform weights. Specifically, we show that it is SQ-hard to distinguish between such a mixture and the standard Gaussian. We further explore how the distribution of weights affects the complexity of this task. Our second main result is a quasi-polynomial upper bound for the aforementioned testing task when most of the weights are uniform while a small fraction of the weights are potentially arbitrary.} }
Endnote
%0 Conference Paper %T On Learning Parallel Pancakes with Mostly Uniform Weights %A Ilias Diakonikolas %A Daniel Kane %A Sushrut Karmalkar %A Jasper C.H. Lee %A Thanasis Pittas %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-diakonikolas25b %I PMLR %P 13601--13621 %U https://proceedings.mlr.press/v267/diakonikolas25b.html %V 267 %X We study the complexity of learning $k$-mixtures of Gaussians ($k$-GMMs) on $\mathbb R^d$. This task is known to have complexity $d^{\Omega(k)}$ in full generality. To circumvent this exponential lower bound on the number of components, research has focused on learning families of GMMs satisfying additional structural properties. A natural assumption posits that the component weights are not exponentially small and that the components have the same unknown covariance. Recent work gave a $d^{O(\log(1/w_{\min}))}$-time algorithm for this class of GMMs, where $w_{\min}$ is the minimum weight. Our first main result is a Statistical Query (SQ) lower bound showing that this quasi-polynomial upper bound is essentially best possible, even for the special case of uniform weights. Specifically, we show that it is SQ-hard to distinguish between such a mixture and the standard Gaussian. We further explore how the distribution of weights affects the complexity of this task. Our second main result is a quasi-polynomial upper bound for the aforementioned testing task when most of the weights are uniform while a small fraction of the weights are potentially arbitrary.
APA
Diakonikolas, I., Kane, D., Karmalkar, S., Lee, J.C. & Pittas, T.. (2025). On Learning Parallel Pancakes with Mostly Uniform Weights. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:13601-13621 Available from https://proceedings.mlr.press/v267/diakonikolas25b.html.

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