CSCI 5525 Machine Learning


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1 CSCI 555 Mchine Lerning
2 Some Deini*ons Qudrtic Form : nn squre mtri R n n : n vector R n the qudrtic orm: It is sclr vlue. We oten implicitly ssume tht is symmetric since / / I we write it s the elements o nd, it is n n i j ij i j
3 Some Deini*ons Qudrtic Form emple d c b d b c d c b d c b
4 Some Deini*ons Positive Deinite PD : symmetric mtri S n For ll nonzero vectors R n, > 0. hen is positive deinite PD Positive Semideinite PSD : symmetric mtri S n For ll vectors R n, 0. hen is positive semideinite PSD Negtive Deinite nd Negtive Semideinite Indeinite 4
5 Some Deini*ons Positive Deinite PD emple
6 Some Deini*ons Eigenvlues nd Eigenvectors : squre mtri R n n λ: C : vector C n I λ, 0, λ is n eigenvlue o nd is the corresponding eigenvector. λ is solution to λi 0. he corresponding eigenvector o λ i is the solution to the liner eqution λ i I 0. here re more eicient methods in prctice to numericlly compute the eigenvlues nd eigenvectors. 6
7 Some Deini*ons Eigenvlues nd Eigenvectors emple, 0 4 I λ λ λ λ λ λ λ 7
8 Proper*es o Eigenvlues nd Eigenvectors he trce o is equl to the sum o its eigenvlues. he determinnt o is equl to the product o its eigenvlues. he rnk o is equl to the number o nonzero eigenvlues o. I is nonsingulr then /λ i is n eigenvlue o with ssocited eigenvector i. he eigenvlues o digonl mtri D digd,... d n re just the digonl entries d,... d n. Digonlizble: We cn write ll the eigenvector equtions together s X XΛ. I the eigenvectors o re linerly independent, XΛX. We sy is digonlizble. 8
9 Eigenvlues nd Eigenvectors o Symmetric Mtrices : symmetric mtri S n ll the eigenvlues o re rel. he eigenvectors o re orthonorml he inner product is 0.. is digonlizble: UΛU Note: U  U UΛU Λy i ll λ i > 0 is positive deinite y ll λ i 0 is positive semideinite n λ y hs both positive nd negtive eigenvlues is indeinite i i 9
10 Wht is Mtri Clculus Clculus Dierentil clculus Derivtive e.g., derivtive unction Integrl clculus Mtri Clculus Etension o clculus to the vector/mtri setting Grdient Hessin 0
11 he Grdient Deinition Function : R m n R : m n mtri he grdient o written s is n m n mtri nd ech element o the mtri is prtil derivtive deined by ij ij
12 he Grdient Emple : mtri clculte ech element o
13 he Grdient Emple he grdient o he generl cse or
14 he Grdient When is vector vector R n wo properties g g For t R, t t wo importnt notes is lwys the sme s the size o the grdient o is deined only i is relvlued unction e.g. we cn t tke the grdient o with respect to the grdient o n! n! 4
15 he Hessin Deinition Function : R n R : n n vector he Hessin mtri with respect to written s is n n n mtri nd ech element o the mtri is prtil derivtive deined by ij i j 5
16 he Hessin Emple : vector 4 clculte ech element o
17 he Hessin Emple he Hessin mtri o In generl, i nd S n, 8 7
18 he Hessin Some notes he Hessin is deined only when is relvlued. Hessin is lwys symmetric. We will only consider tking the Hessin with respect to vector. he Hessin is not the grdient o the grdient. However, the grdient o the ith entry o is the ith column or row o. Some useul results b b i symmetric i symmetric 8
19 pplic*on in Lest Squres Op*miz*on he problem Given ullrnked mtri R m n nd vector b R m Suppose there is no such tht b. Find vector R n, such tht the squre o the Eucliden norm b is minimized. Solve the problem b b b b b b ke the grdient with respect to b b b b b b Set the grdient to zero vector nd we get the solution b b 9
20 0
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