"Learning Spectral Methods by Transformers"

Transformers evolve beyond pattern matching to actually learning mathematical algorithms

Transformers can learn unsupervised algorithms like PCA and clustering through pre-training, enabling them to perform statistical tasks on new data without explicit programming.

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https://arxiv.org/abs/2501.01312

🤔 Original Problem:

→ While Transformers excel at supervised learning tasks, their ability to handle unsupervised learning remains unexplored and lacks theoretical understanding

→ Current research focuses on in-context learning, but doesn't address how Transformers can learn fundamental unsupervised algorithms

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🔍 Solution in this Paper:

→ The paper introduces a multi-layered Transformer that learns spectral methods through pre-training

→ It demonstrates how Transformers can approximate the Power Method algorithm for Principal Component Analysis

→ The architecture uses ReLU attention and averaged multi-head outputs instead of traditional Softmax and concatenated features

→ The model learns to perform both PCA and clustering on Gaussian mixture models without explicit algorithmic programming

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💡 Key Insights:

→ Transformers can learn complex unsupervised algorithms through past experience rather than in-context learning

→ The multi-layer architecture naturally maps to iterative algorithms used in spectral methods

→ Complex algorithms can be broken down into atomic sub-networks within the Transformer

→ The auxiliary matrix design is theoretically important but not necessary in practice

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📊 Results:

→ Achieves 0.95 cosine similarity for top-1 eigenvector prediction

→ Maintains 0.86 accuracy for top-2 and 0.72 for top-3 eigenvectors

→ Performs well on real-world datasets like MNIST with 0.90 accuracy

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