Further taking the usual assumption that X is compact. Let us start with Proposition 3, a central observation needed in Theorem 2. Put into words Now, we can proceed to prove the universality part of Theorem 2. Since the task admits a smooth separator, By Fubini's theorem and Proposition 3, we have F The reader can think of λ as a uniform distribution over G. (as in Theorem 2). The result follows directly from the combination of de Finetti's theorem [ Combining this with Kallenberg's noise transfer theorem we have that the weights and Assumption 1 or ii) is an inner-product decision graph problem as in Definition 3. Further, the task has infinitely (as in Theorem 2). Finally, we follow Proposition 2's proof by simply replacing de Finetti's with Aldous-Hoover's theorem. Define an RLC that samples the linear coefficients as follows.

This research offers a fresh take on neural operators with a framework Representation equivalent Neural Operators (ReNO) designed to address these issues.

Algorithmic stability is a major theme in learning theory, where seminal results have firmly established its close relationship with generalization. Recent research has further highlighted the intricate interplay between stability and additional properties of interest beyond statistical generalization.

Easily integrated, FamO2O statistically enhances existing algorithms' performance.

We also analyze the potentially harmful effects of extrinsic equivariance.