a 1 R N Then y 1 a T 1 x e 1 and b x 1 P P a 1 1 a T 1 P a 1 1 a T 1 P A T y y

# A 1 r n then y 1 a t 1 x e 1 and b x 1 p p a 1 1 a t

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, a 1 R N . Then y 1 = a T 1 x * + e 1 , and b x 1 = P 0 - P 0 a 1 (1 + a T 1 P 0 a 1 ) - 1 a T 1 P 0 ( A T 0 y 0 + y 1 a 1 ) . Set u = P 0 a 1 . Then b x 1 = b x 0 + y 1 u - a T 1 b x 0 1 + a T 1 u u - y 1 · a T 1 u 1 + a T 1 u u = b x 0 + 1 1 + a T 1 u ( y 1 - a T 1 b x 0 ) u . Thus we can update the solution with one vector-matrix multiply (which has cost O ( N 2 )) and two inner products (with cost O ( N )). In addition, we can carry forward the “information matrix” using the update P 1 = P 0 - 1 1 + a T 1 u uu T . In general (for M 1 new measurements), we have b x 1 = P 1 ( A T 0 y 0 + A T 1 y 1 ) = P 1 ( P - 1 0 b x 0 + A T 1 y 1 ) , and since P - 1 0 = P - 1 1 - A T 1 A 1 , 50 Georgia Tech ECE 6250 Fall 2019; Notes by J. Romberg and M. Davenport. Last updated 3:33, November 20, 2019

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this implies b x 1 = P 1 P - 1 1 b x 0 - A T 1 A 1 b x 0 + A T 1 y 1 = b x 0 + K 1 ( y 1 - A 1 b x 0 ) , where K 1 is the “gain matrix” K 1 = P 1 A T 1 . The update for P 1 is P 1 = P 0 - P 0 A T 1 ( I + A 1 P 0 A T 1 ) - 1 A 1 P 0 = P 0 - U ( I + A 1 U ) - 1 U T , where U = P 0 A T 1 is an N × M 1 matrix, and I + A 1 U is M 1 × M 1 . So the cost of the update is O ( M 1 N 2 ) to compute U = P 0 A T 1 , O ( M 2 1 N ) to compute A 1 U , O ( M 3 1 ) to invert 1 ( I + A 1 U ) - 1 , O ( M 2 1 N ) to compute ( I + A 1 U ) - 1 U T , O ( M 1 N 2 ) to take the result of the last step and apply U , O ( N 2 ) to subtract the result of the last step from P 0 . So assuming that M 1 < N , the overall cost is O ( M 1 N 2 ), which is on the order of M 1 vector-matrix multiplies. 1 In practice, it is probably more stable to find and update a factorization of this matrix. But the cost is the same. 51 Georgia Tech ECE 6250 Fall 2019; Notes by J. Romberg and M. Davenport. Last updated 3:33, November 20, 2019
Recursive Least Squares (RLS) Given y 0 = A 0 x * + e 0 y 1 = A 1 x * + e 1 . . . y k = A k x * + e k . . . , RLS is an online algorithm for computing the best estimate for x * from all the measurements it has seen up to the current time. Recursive Least Squares Initialize: ( y 0 appears) P 0 = ( A T 0 A 0 ) - 1 b x 0 = P 0 ( A T 0 y 0 ) for k = 1 , 2 , 3 , . . . do ( y k appears) P k = P k - 1 - P k - 1 A T k ( I + A k P k - 1 A T k ) - 1 A k P k - 1 K k = P k A T k b x k = b x k - 1 + K k ( y k - A k b x k - 1 ) end for 52 Georgia Tech ECE 6250 Fall 2019; Notes by J. Romberg and M. Davenport. Last updated 3:33, November 20, 2019

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