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Hai Pham

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3 papers

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NeurIPS Conference 2021 Conference Paper

Optimal Sketching for Trace Estimation

Shuli Jiang
Hai Pham
David Woodruff
Richard Zhang

Matrix trace estimation is ubiquitous in machine learning applications and has traditionally relied on Hutchinson's method, which requires $O(\log(1/\delta)/\epsilon^2)$ matrix-vector product queries to achieve a $(1 \pm \epsilon)$-multiplicative approximation to $\text{trace}(A)$ with failure probability $\delta$ on positive-semidefinite input matrices $A$. Recently, the Hutch++ algorithm was proposed, which reduces the number of matrix-vector queries from $O(1/\epsilon^2)$ to the optimal $O(1/\epsilon)$, and the algorithm succeeds with constant probability. However, in the high probability setting, the non-adaptive Hutch++ algorithm suffers an extra $O(\sqrt{\log(1/\delta)})$ multiplicative factor in its query complexity. Non-adaptive methods are important, as they correspond to sketching algorithms, which are mergeable, highly parallelizable, and provide low-memory streaming algorithms as well as low-communication distributed protocols. In this work, we close the gap between non-adaptive and adaptive algorithms, showing that even non-adaptive algorithms can achieve $O(\sqrt{\log(1/\delta)}/\epsilon + \log(1/\delta))$ matrix-vector products. In addition, we prove matching lower bounds demonstrating that, up to a $\log \log(1/\delta)$ factor, no further improvement in the dependence on $\delta$ or $\epsilon$ is possible by any non-adaptive algorithm. Finally, our experiments demonstrate the superior performance of our sketch over the adaptive Hutch++ algorithm, which is less parallelizable, as well as over the non-adaptive Hutchinson's method.

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NeurIPS Conference 2020 Conference Paper

Revisiting the Sample Complexity of Sparse Spectrum Approximation of Gaussian Processes

Minh Hoang
Nghia Hoang
Hai Pham
David Woodruff

We introduce a new scalable approximation for Gaussian processes with provable guarantees which holds simultaneously over its entire parameter space. Our approximation is obtained from an improved sample complexity analysis for sparse spectrum Gaussian processes (SSGPs). In particular, our analysis shows that under a certain data disentangling condition, an SSGP's prediction and model evidence (for training) can well-approximate those of a full GP with low sample complexity. We also develop a new auto-encoding algorithm that finds a latent space to disentangle latent input coordinates into well-separated clusters, which is amenable to our sample complexity analysis. We validate our proposed method on several benchmarks with promising results supporting our theoretical analysis.

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AAAI Conference 2019 Conference Paper

Found in Translation: Learning Robust Joint Representations by Cyclic Translations between Modalities

Hai Pham
Paul Pu Liang
Thomas Manzini
Louis-Philippe Morency
Barnabás Póczos

Multimodal sentiment analysis is a core research area that studies speaker sentiment expressed from the language, visual, and acoustic modalities. The central challenge in multimodal learning involves inferring joint representations that can process and relate information from these modalities. However, existing work learns joint representations by requiring all modalities as input and as a result, the learned representations may be sensitive to noisy or missing modalities at test time. With the recent success of sequence to sequence (Seq2Seq) models in machine translation, there is an opportunity to explore new ways of learning joint representations that may not require all input modalities at test time. In this paper, we propose a method to learn robust joint representations by translating between modalities. Our method is based on the key insight that translation from a source to a target modality provides a method of learning joint representations using only the source modality as input. We augment modality translations with a cycle consistency loss to ensure that our joint representations retain maximal information from all modalities. Once our translation model is trained with paired multimodal data, we only need data from the source modality at test time for final sentiment prediction. This ensures that our model remains robust from perturbations or missing information in the other modalities. We train our model with a coupled translationprediction objective and it achieves new state-of-the-art results on multimodal sentiment analysis datasets: CMU-MOSI, ICT- MMMO, and YouTube. Additional experiments show that our model learns increasingly discriminative joint representations with more input modalities while maintaining robustness to missing or perturbed modalities.

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