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Learning Representations from Imperfect Time Series Data via Tensor Rank Regularization
Paul Pu Liang, Zhun Liu, Yao-Hung Hubert Tsai, Qibin Zhao, Ruslan Salakhutdinov, Louis-Philippe Morency
TL;DR
Imperfect multimodal data can be incomplete or noise-corrupted, motivating robust representation learning. The paper uses tensor rank regularization in T2FN to learn multimodal time-series representations, and reports good results across various levels of imperfection.
Problem
Multimodal data may be incomplete or corrupted by noise, making robust representation learning an important research question.
Source: §1 · p. 1Method
T2FN learns multimodal time-series tensor representations and uses tensor rank minimization as a regularizer.
Source: §3.2 · p. 3, §6 · p. 5Results
Experiments on multimodal language data show that T2FN achieves good results across various levels of imperfection.
Source: §6 · p. 5Takeaways & Limitations
The experiments support tensor rank regularization as an approach to robust prediction with imperfect multimodal data.
Source: §6 · p. 5A fuller theoretical analysis is needed; tensor-rank approximation trades computational cost against tightness and may not exist.
Source: §5 · p. 5
Abstract
from arXiv · showhide
There has been an increased interest in multimodal language processing including multimodal dialog, question answering, sentiment analysis, and speech recognition. However, naturally occurring multimodal data is often imperfect as a result of imperfect modalities, missing entries or noise corruption. To address these concerns, we present a regularization method based on tensor rank minimization. Our method is based on the observation that high-dimensional multimodal time series data often exhibit correlations across time and modalities which leads to low-rank tensor representations. However, the presence of noise or incomplete values breaks these correlations and results in tensor representations of higher rank. We design a model to learn such tensor representations and effectively regularize their rank. Experiments on multimodal language data show that our model achieves good results across various levels of imperfection.
1 Introduction
The introduction frames robust representation learning from imperfect multimodal data as a key challenge. It proposes T2FN, which uses tensor rank regularization to model temporal multimodal interactions and their underlying structure.
1 Introduction
- Multimodal language sequences combine heterogeneous language, visual, and acoustic information across tasks such as dialog, question answering, sentiment analysis, and speech recognition.The introduction also gives robot instruction following as an example of integrating verbal instructions, nonverbal behavior, and visual sensing.Source: §1 · p. 1
- Real-world multimodal data may be incomplete because of mismatched modalities or sensor failure, or corrupted by random or structured noise.The resulting research question is how to learn robust representations from such imperfect inputs.Source: §1 · p. 1
- T2FN builds tensor representations of multimodal time series and applies rank regularization without requiring advance identification of imperfect entries or modalities.The model combines temporal nonlinear transformations with regularization of tensor structure.Source: §1 · pp. 1–2
- Clean sequences’ cross-time and cross-modality correlations yield low-rank tensors, while noise or missing values disrupt those correlations and increase rank.Rank minimization is used to learn tensors that better represent the underlying correlations and latent structures.Source: Figure 1 · p. 1, §1 · p. 2
2 Related Work
The related work reviews tensor representations for unimodal and multimodal tasks alongside methods for imperfect multimodal data. It positions this paper’s neural integration of tensor rank minimization as a distinct approach.
Tensor Methods
- Prior tensor methods capture higher-order interactions across time, feature dimensions, and modalities, with applications in both unimodal and multimodal tasks.Examples in the review include language, translation, image captioning, sentiment, speech recognition, and video analysis.Source: §2 · p. 2
Imperfect Data
- Prior imperfect-data approaches include generative models, cascaded residual autoencoders, deep adversarial learning, and translation-based learning.The review notes that these methods often require knowing which entries or modalities are imperfect beforehand.Source: §2 · p. 2
- The paper presents its approach as the first to integrate rank minimization with neural networks for multimodal language data.It aims to combine nonlinear transformations with the mathematical foundations of tensor structures.Source: §2 · p. 2
3 Proposed Method
The method builds temporal multimodal tensor representations and regularizes them using rank-related structure. It motivates this regularization by the increased tensor rank associated with imperfect data.
3 Proposed Method
- The method learns representations from imperfect human-language data across language, visual, and acoustic modalities, using tensor-rank minimization as regularization.The section covers tensor-rank background, T2FN construction, the relationship between imperfection and rank, and the regularizer.Source: §3 · p. 2, §3.4 · p. 3
3.1 Background: Tensor Rank
- Tensor rank is the minimum number of vector-factor terms in an exact CP decomposition; simple outer-product tensors have lower rank than complex tensors.The rank measures how many vectors are required to reconstruct a tensor.Source: §3.1 · pp. 2–3
3.2 Multimodal Tensor Representations
- T2FN extends TFN with temporal LSTM encodings and summed outer products, creating multimodal representations with rank bounded by T rather than TFN’s rank one.Appending a 1 captures unimodal, bimodal, and trimodal interactions.Source: §3.2 · p. 3
3.3 How Does Imperfection Affect Rank?
- Rank is maximal under i.i.d. noise; clean correlations yield lower rank, while noise or missing values raise rank between clean and noisy data.The section says experiments supporting these rank observations appear in §4.2.Source: §3.3 · p. 3
3.4 Tensor Rank Regularization
- Because exact tensor rank and nuclear-norm computation are NP-hard for order ≥3, the model minimizes a scaled Frobenius-norm upper bound alongside prediction loss.The Frobenius-norm term is easily computable and convex, adding negligible computational cost.Source: §3.4 · p. 4
4 Experiments
The experiments examine how imperfect data affects tensor rank and whether rank-regularized T2FN maintains prediction accuracy on CMU-MOSI. Across the tested imperfections, tensor rank increases, while T2FN maintains good performance.
4 Experiments
- The experiments address how imperfection affects T2FN tensor rank and whether rank-regularized T2FN predicts well with imperfect data.The two questions are investigated in Sections 4.2 and 4.3, respectively.Source: §4 · p. 4
4.1 Datasets
- CMU-MOSI contains 2,199 videos annotated for sentiment from −3 to +3, with language, visual, and acoustic features aligned into multimodal sequences.The experiments use train, validation, and test splits and features extracted from GloVe, Facet, and COVAREP.Source: §4.1 · p. 4
4.2 Rank Analysis
- The rank analysis tests clean data, independent entry drops, and modality-specific time-step drops at noise levels from 0.0 to 1.0.Features are dropped during both training and testing; CP decomposition reconstruction error across rank settings provides a rank surrogate because exact tensor rank is NP-hard to determine.Source: §4.2 · p. 4
- Imperfection increases approximate tensor rank on CMU-MOSI, as indicated by reconstruction error.Figure 3(a) shows the rank analysis; the observed shift supports the hypothesis that imperfect data increases tensor rank.Source: §4.2 · pp. 4–5, Figure 3 · p. 5
4.3 Prediction Results
- T2FN with rank regularization maintains good CMU-MOSI test accuracy under imperfect data, with larger improvements on random drop than structured drop.Existing baselines suffer as imperfections increase.Source: §4.3 · p. 5, Figure 3 · p. 5
5 Discussion and Future Work
The discussion identifies open theoretical questions and constraints in tensor rank approximation. It notes trade-offs in bounding rank and the possibility that low-rank approximations do not exist.
5 Discussion and Future Work
- Alternative tensor-rank upper bounds trade computational cost against approximation tightness.Tensor rank can exceed the maximum dimension, and a low-rank approximation may not exist.Source: §5 · p. 5
- The authors identify more thorough theoretical analysis of constructing and regularizing multimodal tensor representations as future work.They describe the rank-regularization method as working well empirically.Source: §5 · p. 5
6 Conclusion
The conclusion presents tensor rank minimization as a way to regularize multimodal time-series representations. It reports good results across varying imperfections in multimodal language data.
6 Conclusion
- The Temporal Tensor Fusion Network learns multimodal tensor representations and regularizes their rank, achieving good results across various levels of imperfection.The conclusion relates low rank in clean sequences to correlations across time and modalities, and higher rank under imperfections to broken correlations.Source: §6 · p. 5