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Let n 2 2 hc an integer . DenoteSn = {(PL,. Pn-1T €R"-I : 0 < Di < for 1 < i<" -L,and 0 < 1 -Pi < 1}.Let (Pi , 'Pr-VT €...

Question

Let n 2 2 hc an integer . DenoteSn = {(PL,. Pn-1T €R"-I : 0 < Di < for 1 < i<" -L,and 0 < 1 -Pi < 1}.Let (Pi , 'Pr-VT € Sn be fixed vector. Consider the Cross Eutropy loss LcE defined OH Sn byLcE (q) =(+2 ) log (-2") 2" log %, where (41, 'n= 41) e Sn, aud the "log" is the natural logarithin (with base Xio(W)-1 ~ 2.71828 . . .) . (10 marks) Find q € Sn such that ~LcE(q) = 0 for <isnm1 (10 marks) For i,j € {1,2. let H;j (4) 8q.89

Let n 2 2 hc an integer . Denote Sn = {(PL,. Pn-1T €R"-I : 0 < Di < for 1 < i<" -L,and 0 < 1 - Pi < 1}. Let (Pi , 'Pr-VT € Sn be fixed vector. Consider the Cross Eutropy loss LcE defined OH Sn by LcE (q) = (+2 ) log (-2") 2" log %, where (41, 'n= 41) e Sn, aud the "log" is the natural logarithin (with base Xio(W)-1 ~ 2.71828 . . .) . (10 marks) Find q € Sn such that ~LcE(q) = 0 for <isnm1 (10 marks) For i,j € {1,2. let H;j (4) 8q.89, LCE(q) . Find Hij(q). Let H(a) (His(a)) = be the (n _ 1) * (n 1) Hessian matrix of LCE- Provc that H(q) positive semni-definite That is, prove that for AHY C1, Gn-1 €R, GCjHis(q) > 0.



Answers

Prove Theorem 8.2 (i) If $A$ has a row (column) of zeros, then $|A|=0$ (ii) If $A$ has two identical rows (columns), then $|A|=0$ (iii) If $A$ is triangular, then $|A|=$ product of diagonal elements. Thus, $|I|=1$ (i) Each term in $|A|$ contains a factor from every row, and so from the row of zeros. Thus, each term of $|A|$ is zero, and so $|A|=0$ (ii) Suppose $1+1 \neq 0$ in $K$. If we interchange the two identical rows of $A$, we still obtain the matrix $A$. Hence, by Problem $8.23,|A|=-|A|,$ and so $|A|=0$ Now suppose $1+1=0$ in $K .$ Then $\operatorname{sgn} \sigma=1$ for every $\sigma \in S_{n} .$ Because $A$ has two identical rows, we can arrange the terms of $A$ into pairs of equal terms. Because each pair is $0,$ the determinant of $A$ is zero. (iii) Suppose $A=\left[a_{i j}\right]$ is lower triangular; that is, the entries above the diagonal are all zero: $a_{i j}=0$ whenever $i<j .$ Consider a term $t$ of the determinant of $A$ : $$t=(\operatorname{sgn} \sigma) a_{1 i_{1}} a_{2 i_{2}} \cdots a_{n i_{n}}, \quad \text { where } \quad \sigma=i_{1} i_{2} \cdots i_{n}$$


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