By Charles L Byrne

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**Example text**

Let yi be the proportion of the total photon count that was recorded at the ith detector. Denote by xj the (unknown) proportion of the total photon count that was emitted from pixel j. Selecting an urn randomly is analogous to selecting which pixel will be the next to emit a photon. Learning the color of the marble is ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ 18 2. Background analogous to learning where the photon was detected; for simplicity we are assuming that all emitted photons are detected, but this is not essential.

X1 ∂xJ It follows from the Cauchy Inequality that |Du f (x)| ≤ ||∇f (x)||2 , with equality if and only if u is parallel to the gradient vector, ∇f (x). The gradient points in the direction of the greatest increase in f (x). 1. Show that the vector a is orthogonal to the hyperplane H = H(a, γ); that is, if u and v are in H, then a is orthogonal to u − v. ˜ is symmetric. 2. 3. Let B be Hermitian. For any x = x1 + ix2 , deﬁne x˜ = (−x2 , x1 )T . Show that the following are equivalent: 1. Bx = λx; ˜x 2.

Given any nonempty closed convex set C and an arbitrary vector x in X , there is a unique member of C closest to x, denoted PC x, the orthogonal (or metric) projection of x onto C. Proof: If x is in C, then PC x = x, so assume that x is not in C. Then d > 0, where d is the distance from x to C. For each positive integer n, select cn in C with ||x − cn ||2 < d + n1 , and ||x − cn ||2 < ||x − cn−1 ||2 . Then the sequence {cn } is bounded; let c∗ be any cluster point. It follows easily that ||x − c∗ ||2 = d and that c∗ is in C.