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 non-zero 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 non-zero eigenvlues o. I is non-singulr 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 rel-vlued 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 rel-vlued. 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 ull-rnked 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

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