Implicit High-Order Moment Tensor Estimation and Learning Latent Variable Models
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arXiv
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| Format: | Preprint |
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2024
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| author | Diakonikolas, Ilias Kane, Daniel M. |
| author_facet | Diakonikolas, Ilias Kane, Daniel M. |
| contents | We study the task of learning latent-variable models. A common algorithmic technique for this task is the method of moments. Unfortunately, moment-based approaches are hampered by the fact that the moment tensors of super-constant degree cannot even be written down in polynomial time. Motivated by such learning applications, we develop a general efficient algorithm for {\em implicit moment tensor computation}. Our framework generalizes the work of~\cite{LL21-opt} which developed an efficient algorithm for the specific moment tensors that arise in clustering mixtures of spherical Gaussians.
By leveraging our implicit moment estimation algorithm, we obtain the first $\mathrm{poly}(d, k)$-time learning algorithms for the following models.
* {\bf Mixtures of Linear Regressions} We give a $\mathrm{poly}(d, k, 1/ε)$-time algorithm for this task, where $ε$ is the desired error.
* {\bf Mixtures of Spherical Gaussians} For density estimation, we give a $\mathrm{poly}(d, k, 1/ε)$-time learning algorithm, where $ε$ is the desired total variation error, under the condition that the means lie in a ball of radius $O(\sqrt{\log k})$. For parameter estimation, we give a $\mathrm{poly}(d, k, 1/ε)$-time algorithm under the {\em optimal} mean separation of $Ω(\log^{1/2}(k/ε))$.
* {\bf Positive Linear Combinations of Non-Linear Activations} We give a general algorithm for this task with complexity $\mathrm{poly}(d, k) g(ε)$, where $ε$ is the desired error and the function $g$ depends on the Hermite concentration of the target class of functions. Specifically, for positive linear combinations of ReLU activations, our algorithm has complexity $\mathrm{poly}(d, k) 2^{\mathrm{poly}(1/ε)}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_15669 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Implicit High-Order Moment Tensor Estimation and Learning Latent Variable Models Diakonikolas, Ilias Kane, Daniel M. Data Structures and Algorithms Machine Learning Statistics Theory We study the task of learning latent-variable models. A common algorithmic technique for this task is the method of moments. Unfortunately, moment-based approaches are hampered by the fact that the moment tensors of super-constant degree cannot even be written down in polynomial time. Motivated by such learning applications, we develop a general efficient algorithm for {\em implicit moment tensor computation}. Our framework generalizes the work of~\cite{LL21-opt} which developed an efficient algorithm for the specific moment tensors that arise in clustering mixtures of spherical Gaussians. By leveraging our implicit moment estimation algorithm, we obtain the first $\mathrm{poly}(d, k)$-time learning algorithms for the following models. * {\bf Mixtures of Linear Regressions} We give a $\mathrm{poly}(d, k, 1/ε)$-time algorithm for this task, where $ε$ is the desired error. * {\bf Mixtures of Spherical Gaussians} For density estimation, we give a $\mathrm{poly}(d, k, 1/ε)$-time learning algorithm, where $ε$ is the desired total variation error, under the condition that the means lie in a ball of radius $O(\sqrt{\log k})$. For parameter estimation, we give a $\mathrm{poly}(d, k, 1/ε)$-time algorithm under the {\em optimal} mean separation of $Ω(\log^{1/2}(k/ε))$. * {\bf Positive Linear Combinations of Non-Linear Activations} We give a general algorithm for this task with complexity $\mathrm{poly}(d, k) g(ε)$, where $ε$ is the desired error and the function $g$ depends on the Hermite concentration of the target class of functions. Specifically, for positive linear combinations of ReLU activations, our algorithm has complexity $\mathrm{poly}(d, k) 2^{\mathrm{poly}(1/ε)}$. |
| title | Implicit High-Order Moment Tensor Estimation and Learning Latent Variable Models |
| topic | Data Structures and Algorithms Machine Learning Statistics Theory |
| url | https://arxiv.org/abs/2411.15669 |