AForster transform is an operation that turns a distribution into one with good anticoncentration properties. While a Forster transform does not always exist, we show that any distribution can be efficiently decomposed as a disjoint mixture of few distributions for which a Forster transform exists and can be computed efficiently. As the main application of this result, we obtain the first polynomial-time algorithm for distribution-independent PAC learning of halfspaces in the Massart noise model with strongly polynomial sample complexity, i.e., independent of the bit complexity of the examples. Previous algorithms for this learning problem incurred sample complexity scaling polynomially with the bit complexity, even though such a dependence is not information-theoretically necessary.

Based on the formulation of the likelihood function pΘ(G|I) = fΘ(G,I)/ZΘ(I), we can reformulate the gradient of log-likelihood function as: ΘL(Θ) = EG pd[ Θ log fΘ(G,I)] Θ log ZΘ(I). Theorem 2. In the initialization phase, the potential function ψtriplet(r,yoh,yot) for modeling label dependency is omitted in p(G|I), yielding a simplified model distribution ˆp(G|I). Now, we can exactly derive that q(G) = ˆp(G|I). Theorem 3. In the update phase, we use the full expression of p(G|I) with the potential function ψtriplet(r,yoh,yot) for modeling label dependency. In this case, maximizing L(q) is equivalent to minimizing the KL divergence term, and the minimum occurs when q(yo) = p(yo,I).

Table 4 contains the full set of topics for the k " 30LDA model introduced in 4.406 Table 4: LDA[6] topic modeling outputs (k=30 topics) when trained on a random sample of documents from mmc4. Topic frequencies are determined by taking the mean distribution over documents in the corpus. Topic names are generated by GPT-4 conditioned on the top 20 words for each topic, prompted by a request for a short 1-2 word summary. Table 5 and Table 6 list the top-50 most frequent top-level domains for documents and images as408 discussed in 4. We show domain statistics in both mmc4and mmc4-core.409 The symbol "*" is employed to denote specific patterns, such as digits or location acronyms, commonly utilized to differentiate sub-sites within the same domain.

This format not only enables few-shot learning via interleaving independent supervised (image, text) examples, but also, more complex prompts involving interaction between images, e.g., "What do image A and image B have in common?" To support this interface, pretraining occurs over web corpora that similarly contain interleaved images+text. To date, however, large-scale data of this form have not been publicly available. We release Multimodal C4 (mmc4), an augmentation of the popular text-only c4 corpus2 with images interleaved. We use a linear assignment algorithm to place images into longer bodies of text using CLIP features [24], a process that we show outperforms alternatives.

Kernel matrix-vector multiplication (KMVM) is one of the most important operations needed in scientific computing with core applications indiffeomorphic registration, geometric learning [11], [31],numerical analysis [28],fluid dynamics [6],and machine learning [27].