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@@ -88,7 +88,7 @@ where $\kappa _ { 0 }$ and $\kappa _ { 1 }$ are given in (4).
 
 In this section, we recall approximation properties of dot-product kernels on the sphere, through spectral decompositions of integral operators in the basis of spherical harmonics. Further background is provided in Appendix A.
 
-Spherical harmonics and description of the RKHS. A standard approach to study the RKHS of a kernel is through the spectral decomposition of an integral operator $T$ given by $\begin{array} { r } { T f ( x ) = \int k ( x , y ) f ( y ) d \tau ( y ) } \end{array}$ for some measure $\tau$ , leading to Mercer’s theorem (e.g., Cucker & Smale, 2002). When inputs lie on the sphere $\mathbb { S } ^ { d - 1 }$ in $d$ dimensions, dot-product kernels of the form $k ( x , x ^ { \prime } ) = \kappa ( x ^ { \prime } x ^ { \prime } )$ are rotationally-invariant, depending only on the angle between $x$ and $x ^ { \prime }$ . Similarly to how translation-invariant kernels are diagonalized in the Fourier basis, rotation-invariant kernels are diagonalized in the basis of spherical harmonics (Smola et al., 2001; Bach, 2017a), which lead to connections between eigenvalue decays and regularity as in the Fourier setting. In particular, if $\tau$ denotes the uniform measure on $\mathbb { S } ^ { d - 1 }$ , then $T Y _ { k , j } = \mu _ { k } Y _ { k , j }$ , where $Y _ { k , j }$ is the $j$ -th spherical harmonic polynomial of degree $k$ , where $k$ plays the role of a frequency as in the Fourier case, and the number of such orthogonal polynomials of degree $k$ is given by $\begin{array} { r } { N ( d , k ) = \frac { 2 k + d - 2 } { k } \binom { k + d - 3 } { d - 2 } } \end{array}$ 2 k+d−3d−2 
, which grows as $k ^ { d - 2 }$ for large $k$ . The eigenvalues $\mu _ { k }$ only depend on the frequency $k$ and are given by
+Spherical harmonics and description of the RKHS. A standard approach to study the RKHS of a kernel is through the spectral decomposition of an integral operator $T$ given by $\begin{array} { r } { T f ( x ) = \int k ( x , y ) f ( y ) d \tau ( y ) } \end{array}$ for some measure $\tau$ , leading to Mercer’s theorem (e.g., Cucker & Smale, 2002). When inputs lie on the sphere $\mathbb { S } ^ { d - 1 }$ in $d$ dimensions, dot-product kernels of the form $k ( x , x ^ { \prime } ) = \kappa ( x ^ { \prime } x ^ { \prime } )$ are rotationally-invariant, depending only on the angle between $x$ and $x ^ { \prime }$ . Similarly to how translation-invariant kernels are diagonalized in the Fourier basis, rotation-invariant kernels are diagonalized in the basis of spherical harmonics (Smola et al., 2001; Bach, 2017a), which lead to connections between eigenvalue decays and regularity as in the Fourier setting. In particular, if $\tau$ denotes the uniform measure on $\mathbb { S } ^ { d - 1 }$ , then $T Y _ { k , j } = \mu _ { k } Y _ { k , j }$ , where $Y _ { k , j }$ is the $j$ -th spherical harmonic polynomial of degree $k$ , where $k$ plays the role of a frequency as in the Fourier case, and the number of such orthogonal polynomials of degree $k$ is given by $\begin{array} { r } { N ( d , k ) = \frac { 2 k + d - 2 } { k } \binom { k + d - 3 } { d - 2 } } \end{array}$ 2 k+d−3d−2 
, which grows as $k ^ { d - 2 }$ for large $k$ . The eigenvalues $\mu _ { k }$ only depend on the frequency $k$ and are given by
 
 $$
 \mu _ { k } = \frac { \omega _ { d - 2 } } { \omega _ { d - 1 } } \int _ { - 1 } ^ { 1 } \kappa ( t ) P _ { k } ( t ) ( 1 - t ^ { 2 } ) ^ { ( d - 3 ) / 2 } d t ,
@@ -317,7 +317,7 @@ Difan Zou, Yuan Cao, Dongruo Zhou, and Quanquan Gu. Stochastic gradient descent
 
 In this section, we provide some background on spherical harmonics needed for our study of approximation. See (Efthimiou & Frye, 2014; Atkinson & Han, 2012; Ismail, 2005) for references, as well as (Bach, 2017a, Appendix D). We consider inputs on the $d - 1$ sphere $\mathbb { S } ^ { d - 1 } = \{ x \in \mathbb { R } ^ { d } , \| x \| = 1 \}$ .
 
-We recall some properties of the spherical harmonics $Y _ { k , j }$ introduced in Section 2.2. For j = 1, . . . , N (d, k), where N (d, k) = 2k+d−2 k+d−3
, the spherical harmonics $Y _ { k , j }$ are homogeneous harmonic polynomials of degree $k$ that are orthonormal with respect to the uniform distribution $\tau$ on the $d$ –1 sphere. The degree $k$ plays the role of an integer frequency, as in Fourier series, and the collection $\{ Y _ { k , j } , k \geq 0 , j = 1 , \ldots , N ( d , k ) \}$ forms an orthonormal basis of $L ^ { 2 } ( \mathbb { S } ^ { d - 1 } , d \tau )$ . As with Fourier series, there are tight connections between decay of coefficients in this basis w.r.t. $k$ , and regularity/differentiability of functions, in this case differentiability on the sphere. This follows from the fact that spherical harmonics are eigenfunctions of the Laplace-Beltrami operator on the sphere $\Delta _ { \mathbb { S } ^ { d - 1 } }$ (see Efthimiou & Frye, 2014, Proposition 4.5):
+We recall some properties of the spherical harmonics $Y _ { k , j }$ introduced in Section 2.2. For j = 1, . . . , N (d, k), where N (d, k) = 2k+d−2 k+d−3
, the spherical harmonics $Y _ { k , j }$ are homogeneous harmonic polynomials of degree $k$ that are orthonormal with respect to the uniform distribution $\tau$ on the $d$ –1 sphere. The degree $k$ plays the role of an integer frequency, as in Fourier series, and the collection $\{ Y _ { k , j } , k \geq 0 , j = 1 , \ldots , N ( d , k ) \}$ forms an orthonormal basis of $L ^ { 2 } ( \mathbb { S } ^ { d - 1 } , d \tau )$ . As with Fourier series, there are tight connections between decay of coefficients in this basis w.r.t. $k$ , and regularity/differentiability of functions, in this case differentiability on the sphere. This follows from the fact that spherical harmonics are eigenfunctions of the Laplace-Beltrami operator on the sphere $\Delta _ { \mathbb { S } ^ { d - 1 } }$ (see Efthimiou & Frye, 2014, Proposition 4.5):
 
 $$
 \Delta _ { \mathbb { S } ^ { d - 1 } } Y _ { k , j } = - k ( k + d - 2 ) Y _ { k , j } .