Neural Information Processing Systems http://nips.cc/

Information Pursuit (IP) is a classical active testing algorithm for predicting an output by sequentially and greedily querying the input in order of information gain. However, IP is computationally intensive since it involves estimating mutual information in high-dimensional spaces. This paper explores Orthogonal Matching Pursuit (OMP) as an alternative to IP for greedily selecting the queries. OMP is a classical signal processing algorithm for sequentially encoding a signal in terms of dictionary atoms chosen in order of correlation gain. In each iteration, OMP selects the atom that is most correlated with the signal residual (the signal minus its reconstruction thus far).

This paper studies the problem of sparse regression where the goal is to learn a sparse vector that best optimizes a given objective function. Under the assumption that the objective function satisfies restricted strong convexity (RSC), we analyze orthogonal matching pursuit (OMP), a greedy algorithm that is used heavily in applications, and obtain support recovery result as well as a tight generalization error bound for OMP. Furthermore, we obtain lower bounds for OMP, showing that both our results on support recovery and generalization error are tight up to logarithmic factors. To the best of our knowledge, these support recovery and generalization bounds are the first such matching upper and lower bounds (up to logarithmic factors) for {\em any} sparse regression algorithm under the RSC assumption.

We study the problem of sparse regression where the goal is to learn a sparse vector that best optimizes a given objective function.

We study quantum sparse recovery in non-orthogonal, overcomplete dictionaries: given coherent quantum access to a state and a dictionary of vectors, the goal is to reconstruct the state up to $\ell_2$ error using as few vectors as possible. We first show that the general recovery problem is NP-hard, ruling out efficient exact algorithms in full generality. To overcome this, we introduce Quantum Orthogonal Matching Pursuit (QOMP), the first quantum analogue of the classical OMP greedy algorithm. QOMP combines quantum subroutines for inner product estimation, maximum finding, and block-encoded projections with an error-resetting design that avoids iteration-to-iteration error accumulation. Under standard mutual incoherence and well-conditioned sparsity assumptions, QOMP provably recovers the exact support of a $K$-sparse state in polynomial time. As an application, we give the first framework for sparse quantum tomography with non-orthogonal dictionaries in $\ell_2$ norm, achieving query complexity $\widetilde{O}(\sqrt{N}/ε)$ in favorable regimes and reducing tomography to estimating only $K$ coefficients instead of $N$ amplitudes. In particular, for pure-state tomography with $m=O(N)$ dictionary vectors and sparsity $K=\widetilde{O}(1)$ on a well-conditioned subdictionary, this circumvents the $\widetildeΩ(N/ε)$ lower bound that holds in the dense, orthonormal-dictionary setting, without contradiction, by leveraging sparsity together with non-orthogonality. Beyond tomography, we analyze QOMP in the QRAM model, where it yields polynomial speedups over classical OMP implementations, and provide a quantum algorithm to estimate the mutual incoherence of a dictionary of $m$ vectors in $O(m/ε)$ queries, improving over both deterministic and quantum-inspired classical methods.

Many signals, such as spike trains recorded in multi-channel electrophysiological recordings, may be represented as the sparse sum of translated and scaled copies of waveforms whose timing and amplitudes are of interest. From the aggregate signal, one may seek to estimate the identities, amplitudes, and translations of the waveforms that compose the signal. Here we present a fast method for recovering these identities, amplitudes, and translations. The method involves greedily selecting component waveforms and then refining estimates of their amplitudes and translations, moving iteratively between these steps in a process analogous to the well-known Orthogonal Matching Pursuit (OMP) algorithm. Our approach for modeling translations borrows from Continuous Basis Pursuit (CBP), which we extend in several ways: by selecting a subspace that optimally captures translated copies of the waveforms, replacing the convex optimization problem with a greedy approach, and moving to the Fourier domain to more precisely estimate time shifts. We test the resulting method, which we call Continuous Orthogonal Matching Pursuit (COMP), on simulated and neural data, where it shows gains over CBP in both speed and accuracy.

We present a training-free method to transplant tokenizers in pretrained large language models (LLMs) by reconstructing unseen token embeddings via Orthogonal Matching Pursuit (OMP). Specifically, we approximate each out-of-vocabulary token as a sparse linear combination of shared tokens, in two phases: first, compute each new token's representation in the donor embedding space with a small dictionary of shared anchor tokens, then transfer these same sparse coefficients back into the base model's embedding space. On two challenging cross-tokenizer tasks--Llama$\to$Mistral NeMo (12B) and Qwen$\to$Llama (1B)--we show that OMP achieves best zero-shot preservation of the base model's performance across multiple benchmarks, while other zero-shot approaches degrade significantly. Compared to baselines (zero-init, mean-init, and existing approaches like WECHSEL, FOCUS, ZETT), OMP consistently achieves the best overall performance, effectively bridging large tokenizer discrepancies without gradient updates. Our analysis further identifies mismatched numerical tokenization schemes as a critical challenge for preserving mathematical reasoning capabilities. This technique enables direct reuse of pretrained model weights with new tokenizers, facilitating cross-tokenizer knowledge distillation, speculative decoding, ensembling, merging, and domain-specific vocabulary adaptations. We integrate our method into the open-source mergekit-tokensurgeon tool for post hoc vocabulary realignment.

Many signals, such as spike trains recorded in multi-channel electrophysiological recordings, may be represented as the sparse sum of translated and scaled copies of waveforms whose timing and amplitudes are of interest. From the aggregate signal, one may seek to estimate the identities, amplitudes, and translations of the waveforms that compose the signal. Here we present a fast method for recovering these identities, amplitudes, and translations. The method involves greedily selecting component waveforms and then refining estimates of their amplitudes and translations, moving iteratively between these steps in a process analogous to the well-known Orthogonal Matching Pursuit (OMP) algorithm [11]. Our approach for modeling translations borrows from Continuous Basis Pursuit (CBP) [4], which we extend in several ways: by selecting a subspace that optimally captures translated copies of the waveforms, replacing the convex optimization problem with a greedy approach, and moving to the Fourier domain to more precisely estimate time shifts. We test the resulting method, which we call Continuous Orthogonal Matching Pursuit (COMP), on simulated and neural data, where it shows gains over CBP in both speed and accuracy.

Many signals, such as spike trains recorded in multi-channel electrophysiological recordings, may be represented as the sparse sum of translated and scaled copies of waveforms whose timing and amplitudes are of interest. From the aggregate signal, one may seek to estimate the identities, amplitudes, and translations of the waveforms that compose the signal. Here we present a fast method for recovering these identities, amplitudes, and translations. The method involves greedily selecting component waveforms and then refining estimates of their amplitudes and translations, moving iteratively between these steps in a process analogous to the well-known Orthogonal Matching Pursuit (OMP) algorithm. Our approach for modeling translations borrows from Continuous Basis Pursuit (CBP), which we extend in several ways: by selecting a subspace that optimally captures translated copies of the waveforms, replacing the convex optimization problem with a greedy approach, and moving to the Fourier domain to more precisely estimate time shifts.

Information Pursuit (IP) is a classical active testing algorithm for predicting an output by sequentially and greedily querying the input in order of information gain. However, IP is computationally intensive since it involves estimating mutual information in high-dimensional spaces. This paper explores Orthogonal Matching Pursuit (OMP) as an alternative to IP for greedily selecting the queries. OMP is a classical signal processing algorithm for sequentially encoding a signal in terms of dictionary atoms chosen in order of correlation gain. In each iteration, OMP selects the atom that is most correlated with the signal residual (the signal minus its reconstruction thus far).