Original: Assume ω ℓ is an eigenvalue of P associated with the eigenspace that is approximately spanned by { Y ℓ,p } N ( d,ℓ ) p =1 .
Revised: Assume ω ℓ is an eigenvalue of P associated with the eigenspace that is approximately spanned by { Y ℓ,p } N ( d,ℓ ) p =1 .

Original: Note that this is the case
Revised: Note that this is the case

Original: when P =
Revised: when P =

Original: P
Revised: P

Original: s is associated with the squared- H s loss on
Revised: s is associated with the squared- H s loss on

Original: S d − 1 .
Revised: S d − 1 .

Original: Then, by eq.
Revised: Then, by eq.

Original: (88), we have
Revised: (88), we have

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: where ˆ y ℓ,p = (ˆ y ℓ,p 1 , . .
Revised: where ˆ y ℓ,p = (ˆ y ℓ,p 1 , . .

Original: , ˆ y ℓ,pN ) is defined by the decomposition
Revised: , ˆ y ℓ,pN ) is defined by the decomposition

Original: [Equation]
Revised: [Equation]

Original: Equation (90) demonstrates that if ω ℓ is relatively large for bigger ℓ (e.g., when we use the squaredH s loss with a large s > 0 ), then we expect that γ ℓ,p ( k ) decays much faster for high frequencies than the low ones, and vice versa.
Revised: Equation (90) demonstrates that if ω ℓ is relatively large for bigger ℓ (e.g., when we use the squaredH s loss with a large s > 0 ), then we expect that γ ℓ,p ( k ) decays much faster for high frequencies than the low ones, and vice versa.

Original: This connects frequency bias in operator learning to the spectral properties of P s .
Revised: This connects frequency bias in operator learning to the spectral properties of P s .

Original: We studied and justified the phenomena in the autoencoder experiment in Sec.
Revised: We studied and justified the phenomena in the autoencoder experiment in Sec.

Original: H C OMPUTATION OF Q UADRATURE W EIGHTS
Revised: H C OMPUTATION OF Q UADRATURE W EIGHTS

Original: In practice, the training dataset usually does not come with a carefully designed quadrature rule.
Revised: In practice, the training dataset usually does not come with a carefully designed quadrature rule.

Original: Hence, we inevitably need to compute a set of quadrature weights before training the NN.
Revised: Hence, we inevitably need to compute a set of quadrature weights before training the NN.

Original: In this section, we briefly discuss methods for computing positive quadrature weights.
Revised: In this section, we briefly discuss methods for computing positive quadrature weights.

Original: Given a set of points { x i } n i =1 on S d − 1 , we wish to construct a quadrature rule so that
Revised: Given a set of points { x i } n i =1 on S d − 1 , we wish to construct a quadrature rule so that

Original: [Equation]
Revised: [Equation]

Original: for sufficiently smooth f , where c i > 0 are positive quadrature weights.
Revised: for sufficiently smooth f , where c i > 0 are positive quadrature weights.

Original: One approach that could give us a very accurate quadrature rule is to guarantee that
Revised: One approach that could give us a very accurate quadrature rule is to guarantee that

Original: [Equation]
Revised: [Equation]

Original: where dim (Π d ℓ ) ≤ n .
Revised: where dim (Π d ℓ ) ≤ n .

Original: The one-dimensional case of such quadrature rules was studied in (Austin & Trefethen, 2017; Yu & Townsend, 2022) and the general higher-dimensional case was analyzed in (Mhaskar et al., 2000; Dai & Xu, 2013).
Revised: The one-dimensional case of such quadrature rules was studied in (Austin & Trefethen, 2017; Yu & Townsend, 2022) and the general higher-dimensional case was analyzed in (Mhaskar et al., 2000; Dai & Xu, 2013).

Original: Given any dataset { x i } ni =1 and ℓ so that dim (Π dℓ ) ≤ n , one cannot guarantee the existence of a positive quadrature rule satisfying eq.
Revised: Given any dataset { x i } ni =1 and ℓ so that dim (Π dℓ ) ≤ n , one cannot guarantee the existence of a positive quadrature rule satisfying eq.

Original: (91) (Mhaskar et al., 2000; Dai & Xu, 2013), even when the distribution of { x i } ni =1 is very regular (Yu & Townsend, 2022).
Revised: (91) (Mhaskar et al., 2000; Dai & Xu, 2013), even when the distribution of { x i } ni =1 is very regular (Yu & Townsend, 2022).

Original: On the other hand, by choosing ℓ to be small, we can eventually find an ℓ for which eq.
Revised: On the other hand, by choosing ℓ to be small, we can eventually find an ℓ for which eq.

Original: (91) holds.
Revised: (91) holds.

Original: When such a positive quadrature rule exists, Mhaskar et al.
Revised: When such a positive quadrature rule exists, Mhaskar et al.

Original: (2000) proposed to solve the following
Revised: (2000) proposed to solve the following

Original: quadratic
Revised: quadratic

Original: program
Revised: program

Original: [Equation]
Revised: [Equation]

Original: While eq.
Revised: While eq.

Original: (91) gives us guarantee on the accuracy of the quadrature rule (provided ℓ is not too small), it is not always practical to compute the quadrature weights in this way.
Revised: (91) gives us guarantee on the accuracy of the quadrature rule (provided ℓ is not too small), it is not always practical to compute the quadrature weights in this way.

Original: Indeed, if d is large, then we need tremendous amount of points to guarantee that dim (Π dℓ ) ≤ n even for a small ℓ .
Revised: Indeed, if d is large, then we need tremendous amount of points to guarantee that dim (Π dℓ ) ≤ n even for a small ℓ .

Original: Also, if we have too many data points, then the quadratic program can get infeasible to solve.
Revised: Also, if we have too many data points, then the quadratic program can get infeasible to solve.

Original: Hence, we need some other methods for computing quadrature weights that, albeit less accurate, can be applied more cheaply to general datasets.
Revised: Hence, we need some other methods for computing quadrature weights that, albeit less accurate, can be applied more cheaply to general datasets.

Original: One of the many possible approaches is to do kernel density estimation (Rosenblatt, 1956; Parzen, 1962).
Revised: One of the many possible approaches is to do kernel density estimation (Rosenblatt, 1956; Parzen, 1962).

Original: To do so, we fix a positive kernel K ( x , y ) =
Revised: To do so, we fix a positive kernel K ( x , y ) =

Original: K (arccos( x ⊤ y )) defined on S d − 1 × S d − 1 .
Revised: K (arccos( x ⊤ y )) defined on S d − 1 × S d − 1 .

Original: A common choice of K can be the Gaussian density function of standard deviation 1 centered at 0 .
Revised: A common choice of K can be the Gaussian density function of standard deviation 1 centered at 0 .

Original: For each h > 0 , we then define a function p h on S d −by
Revised: For each h > 0 , we then define a function p h on S d −by

Original: [Equation]
Revised: [Equation]

Original: The bandwidth h is a hyperparameter, and with an appropriate h , the function p h is an (unnormalized) estimate of the density function of the distribution of nodes.
Revised: The bandwidth h is a hyperparameter, and with an appropriate h , the function p h is an (unnormalized) estimate of the density function of the distribution of nodes.

Original: Hence, by setting
Revised: Hence, by setting

Original: [Equation]
Revised: [Equation]

Original: we obtain a positive quadrature rule that approximates the integral of smooth functions on S d − 1 .
Revised: we obtain a positive quadrature rule that approximates the integral of smooth functions on S d − 1 .