Designing Information Devices and Systems I Spring 2018 Lecture Notes Note 25

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1 EECS 6 Designing Information Devices and Systems I Spring 8 Lecture Notes Note 5 5. Speeding up OMP In the last lecture note, we introduced orthogonal matching pursuit OMP, an algorithm that can extract information from sparse signals. Recall that in each iteration of the algorithm, first we found one on device from the strongest signal component. hen we computed the projection of the measurement vector onto the subspace spanned by the songs from on devices found so far. Finally, we calculated the residual and used this to look for the next strongest song in the next iteration. If we let j be a matrix whose columns are the songs found so far in iteration j and let be the received signal, we compute j j j j to get the projection of onto span j. Matrix inversion and matrix multiplication can be computationally expensive - inversion is On 3 and multiplication is On. Is there a way to avoid doing such computations? Yes! It turns out that if the columns of j are mutually orthogonal to each other, the projection of onto span j is the sum of the projection of onto each of the columns of j. Recall that the projection of a vector on to any other nonzero vector b of the same size is b = b b b. his is fast to compute since computing the dot product of two vectors and the norm of a vector is linear in time in the number of components in the vectors. Let s take a look at the case where j = and the signatures are mutually orthogonal. Suppose the songs found so far are S and, i.e., = S. Since S and are orthogonal to each other, we have S =. he least squares solution of x = is given by: ˆ x =. and we now multiply by to obtain the projection of the least sqaures solution onto the subspace spanned by : =. 3 EECS 6, Spring 8, Note 5

2 Let s first compute the term : ] S = S 4 S = S S ] S S S 5 S = S. 6 7 hus, we have = S. 8 S hen the projection of onto span is = 9 ] = S S S S S ] = S S S S S S = S S S S = S + S S S. 3 S Observe that the first term in the sum above is the projection of onto S and the second term is the projection of onto. Generalizing, the projection of onto span n where n has mutually orthogonal columns is S n = S + S S S + + S n S S n. 4 S n Furthermore, observe that if S,..., S n are unit vectors i.e., they all have length, then the above further reduces to n = S S + S S + + S n S n, 5 EECS 6, Spring 8, Note 5

3 which further reduces our computation. We see that we can speed up OMP considerably if the songs S,..., S n are mutually orthogonal to each other and are of unit length. Note: In this case, we cannot orthnormalize LL of the songs S,... S n because there are more songs than the length of the song and therefore they cannot all be linearly independent. However, since we are dealing with a sparse system, in our example at most devices transmit at once, we only have to orthonormalize some of the vectors. Now the question is how do we convert any set of linearly independent vectors to a set of mutually orthogonal unit vectors that span the same vector space? We will answer this in the next section. 5. Gram-Schmidt Process Before we begin, let s remind ourselves that the following subspaces are equivalent for any pairs of linearly independent vectors v, v : span v, v span v, α v span v, v + v span v, v v span v, v α v Now what should α be if we would like v and v α v to be orthogonal to each other? We want α v to be the projection of v onto v. Let s solve this algebraically using the definition of orthogonality: v v α v = 6 v v α v = 7 α = v v v 8 Definition 5. Orthonormal: set of vectors { S,..., S n } is orthonormal if all the vectors are mutually orthogonal to each other and all are of unit length. Gram Schmidt is an algorithm that takes a set of linearly independent vectors { S,..., S n } and generates an orthonormal set of vectors {q,...,q n } that span the same vector space as the original set. Concretely, {q,...,q n } needs to satisfy the following: span{s,...,s n } = span{q,...,q n } {q,...,q n } is an orthonormal set of vectors Now let s see how we can do this with a set of three vectors { S,, S 3 } that is linearly independent of each other. EECS 6, Spring 8, Note 5 3

4 Step : Find unit vector q such that span{ q } = span{ S }. Since span{ S } is a one dimensional vector space, the unit vector that span the same vector space would just be the normalized vector point at the same direction as S. We have q = S. 9 S Step : Given q from the previous step, find q such that span{ q, q } = span{ S, } and orthogonal to q. We know that the projection of on q would be orthogonal to q as seen earlier. Hence, a vector e orthogonal to q where span{ q, e } = span{ S, } is e = S q q. Normalizing, we have q = e e. Step 3: Now given q and q in the previous steps, we would like to find q 3 such that span{ q, q, q 3 } = span{ S,, S 3 }. We know that the projection of S 3 onto the subspace spanned by q, q is S3 q q + S3 q q. We know that e 3 = S 3 S3 q q S3 q q is orthogonal to q and q. Normalizing, we have q 3 = e 3 e 3. We can generalize the above procedure for any number of linearly independent vectors as follows: Inputs set of linearly independent vectors { S,..., S n }. Outputs n orthonormal set of vectors { q,..., q n }, where span{ S,..., S n } = span{ q,..., q n }. Gram Schmidt Procedure compute q : q = S S for i =...n:. Compute the vector e i, such that span{ q,..., e i } = span{ S,..., S i }: e i = S i i j= Si q j q j. Normalize to compute q i : q i = e i e i EECS 6, Spring 8, Note 5 4

5 5.3 Implementing Gram Schmidt for OMP Now we would like to use Gram Schmidt to speed up OMP. Recall the step in OMP that augments matrix with the newest version of the song: = S Ni i ] and then uses least squares to obtain the message value: x =. We now know that with each iteration, that computation gets harder and harder. herefore, let us now build a matrix Q containing orthonormal vectors to use as a subspace rather than. We can initialize Q = ] and once we identify a song, we can then perform Gram Schmidt on Q and the song. In this case the received signal is x and the residual is. Procedure: Initialize the following values: = x, j =, k =, F = {/}, Q = ], b = while j k & th :. Cross correlate with the shifted versions of all songs. Find the song index, i, and the shifted version of the song, S Ni i, with which the received signal has the highest correlation value.. Set v j = S Ni i 3. dd i to the set of song indices, F. 4. Perform Gram Schmidt on Q and v j a Find e j which is v j projection of v j onto Q : e j = v j v j q q v j q j q j b Find q j : q j = e j e j c Column concatenate matrix Q with q j : Q = Q q j ] 5. Now that vectors are orthonormal, can simply project received signal onto newest column and add: b j = b j + x q j q j 6. Update the residual value by subtracting: = x b j 7. Update the counter: j = j + EECS 6, Spring 8, Note 5 5