Nyströmformer is an efficient Transformer architecture that approximates the full softmax attention matrix using the Nyström method—a classical technique for approximating large kernel matrices by sampling a subset of landmark points and reconstructing the full matrix from this subset. Nyströmformer selects m landmark tokens (via segment-means or learned selection) and uses them to approximate the N×N attention matrix as a product of three smaller matrices, achieving O(N·m) complexity.

Why Nyströmformer Matters in AI/ML: Nyströmformer provides high-quality attention approximation that preserves the softmax attention's properties more faithfully than linear attention or random feature methods, achieving near-exact attention quality with significantly reduced computational cost.

• Nyström approximation — The full attention matrix A = softmax(QK^T/√d) is approximated as à = A_{NM} · A_{MM}^{-1} · A_{MN}, where M is the set of m landmark tokens, A_{NM} is the N×m attention between all tokens and landmarks, and A_{MM} is the m×m attention among landmarks • Landmark selection — The m landmark tokens are selected by averaging consecutive segments of the sequence: each landmark represents the mean of N/m consecutive tokens, providing a uniform coverage of the sequence; this is simpler than random sampling and provides consistent quality • Pseudo-inverse stability — Computing A_{MM}^{-1} requires inverting an m×m matrix, which can be numerically unstable; Nyströmformer uses iterative methods (Newton's method for matrix inverse) to compute a stable pseudo-inverse without explicit matrix inversion • Approximation quality — With m=64-256 landmarks, Nyströmformer achieves 99%+ of full attention quality on standard NLP benchmarks, outperforming Performer, Linformer, and other efficient attention methods on long-range tasks • Complexity analysis — Computing A_{NM} costs O(N·m·d), A_{MM}^{-1} costs O(m³), and the full approximation costs O(N·m·d + m³); for m << N, this is effectively O(N·m·d), linear in sequence length

Nyströmformer brings the classical Nyström matrix approximation method to Transformers, providing one of the highest-quality efficient attention approximations through landmark-based reconstruction that faithfully preserves softmax attention patterns while reducing quadratic complexity to linear, achieving the best quality-efficiency tradeoff among efficient attention methods.