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@@ -156,7 +156,7 @@ Whereas the $( \alpha , \theta )$ -diatomic LJSDs explicitly enforce the trace n
 
 Deriving conditions on whether a shift will hurt or help a model’s performance is crucial to building an understanding of covariate shift. Motivated in part by the above results for linear regression, and in part by the results for random feature regression that we present in $\mathrm { S e c . } \underline { {   \bar { 5 . 3 }  } }$ we introduce the following definition of shift strength, which is a direct generalization of the conditions of Prop. 3.2:
 
-Definition 4.1. Let $\mu _ { 1 }$ and $\mu _ { 2 }$ be LJSDs with the same marginal distribution of $\lambda ,$ , denoted $\mu _ { t r a i n }$ . If the asymptotic overlap coefficients are such that $\mathbb { E } _ { \mu _ { 1 } } [ r | \lambda ] / \mathbb { E } _ { \mu _ { 2 } } [ r | \lambda ]$ is nondecreasing as a function of $\lambda$ on the support of $\mu _ { t r a i n }$ and $\mathbb { E } _ { \mu _ { 1 } } [ r ] \le \mathbb { E } _ { \mu _ { 2 } } [ r ]$ , we say $\mu _ { 1 }$ is easier than $\mu _ { 2 }$ (or $\mu _ { 2 }$ is harder than $\mu _ { 1 . }$ ), and write $\mu _ { 1 } \leq \mu _ { 2 }$ . Comparing against the case of no shift $\mu _ { \emptyset }$ , we say $\mu _ { 1 }$ is easy when $\mu _ { 1 } \leq \mu _ { \emptyset }$ and hard when µ1  µ .
+Definition 4.1. Let $\mu _ { 1 }$ and $\mu _ { 2 }$ be LJSDs with the same marginal distribution of $\lambda ,$ , denoted $\mu _ { t r a i n }$ . If the asymptotic overlap coefficients are such that $\mathbb { E } _ { \mu _ { 1 } } [ r | \lambda ] / \mathbb { E } _ { \mu _ { 2 } } [ r | \lambda ]$ is nondecreasing as a function of $\lambda$ on the support of $\mu _ { t r a i n }$ and $\mathbb { E } _ { \mu _ { 1 } } [ r ] \le \mathbb { E } _ { \mu _ { 2 } } [ r ]$ , we say $\mu _ { 1 }$ is easier than $\mu _ { 2 }$ (or $\mu _ { 2 }$ is harder than $\mu _ { 1 . }$ ), and write $\mu _ { 1 } \leq \mu _ { 2 }$ . Comparing against the case of no shift $\mu _ { \emptyset }$ , we say $\mu _ { 1 }$ is easy when $\mu _ { 1 } \leq \mu _ { \emptyset }$ and hard when µ1  µ .
 
 A priori, there is little reason to hope that such a model-independent definition of shift strength would adequately characterize a shift’s impact on the total error, bias, or variance of a given model. Even for the relatively simple case of random feature kernel regression, the nonlinear feature maps of Eq. $\mathbb { Q } )$ would seem to inextricably couple the covariance distribution to the model.
 
@@ -193,7 +193,7 @@ $$
 \begin{array} { l } { { \displaystyle B _ { \mu } = \phi \mathcal { T } _ { 1 , 2 } ^ { * } } } \\ { { \displaystyle V _ { \mu } = - \rho \frac { \psi } { \phi } \frac { \partial x } { \partial \gamma } \bigg ( \mathcal { T } _ { 1 , 1 } ( \omega + \phi \mathcal { T } _ { 1 , 2 } ) ( \omega + \mathcal { T } _ { 1 , 1 } ^ { * } ) + \frac { \phi ^ { 2 } } { \psi } \gamma \bar { \tau } \mathcal { T } _ { 1 , 2 } \mathcal { T } _ { 2 , 2 } ^ { * } } } \\ { { \displaystyle \quad \quad + \gamma \tau \mathcal { T } _ { 2 , 2 } ( \omega + \phi \mathcal { T } _ { 1 , 2 } ^ { * } ) + \sigma _ { \varepsilon } ^ { 2 } \Big ( ( \omega + \phi \mathcal { T } _ { 1 , 2 } ) ( \omega + \mathcal { T } _ { 1 , 1 } ^ { * } ) + \frac { \phi } { \psi } \gamma \bar { \tau } \mathcal { T } _ { 2 , 2 } ^ { * } \Big ) \bigg ) , } } \end{array}
 $$
 
-where x is the unique nonnegative real root of x = 1⌧!+I1,1 , @x@ $\begin{array} { r } { \frac { 1 - \gamma \tau } { \omega + \mathcal { T } _ { 1 , 1 } } , \frac { \partial x } { \partial \gamma } = - \frac { x } { \gamma + \rho \gamma ( \tau \psi / \phi + \bar { \tau } ) ( \omega + \phi \mathcal { T } _ { 1 , 2 } ) } } \end{array}$ , and
+where x is the unique nonnegative real root of x = 1⌧!+I1,1 , @x@ $\begin{array} { r } { \frac { 1 - \gamma \tau } { \omega + \mathcal { T } _ { 1 , 1 } } , \frac { \partial x } { \partial \gamma } = - \frac { x } { \gamma + \rho \gamma ( \tau \psi / \phi + \bar { \tau } ) ( \omega + \phi \mathcal { T } _ { 1 , 2 } ) } } \end{array}$ , and
 
 $$
 \tau = \frac { \sqrt { ( \psi - \phi ) ^ { 2 } + 4 x \psi \phi \gamma / \rho } + \psi - \phi } { 2 \psi \gamma } a n d \bar { \tau } = \frac { 1 } { \gamma } + \frac { \psi } { \phi } \bigl ( \tau - \frac { 1 } { \gamma } \bigr ) .