DsRec公式

未分类

---

在此输入正文

$$Pr[S_m \geq \epsilon] \leq e^{-t \epsilon} E[e^{t S_m}]= E\left [ e^{tS_{m-1}}E \left [e^{tV_m}|X_1,\ldots,X_m \right] \right]$$

$$\leq E\left [ e^{tS_{m-1}} {e^{t^2 {S_m}^2 /8}}\right ]$$
$$\mathop {\min }\limits_{{f_k} \in H,{x_s}} {\sum\limits_{i = 1}^l {\sum\limits_{k = 1}^c {\left( {{y_{ik}} - {f_k}\left( {{x_i}} \right)} \right)} } ^2} + \sum\limits_{k = 1}^c {\left( {\mathop y\limits^ \wedge _{sk}^2 + 2\left| {{f_k}\left( {{x_s}} \right)} \right|} \right)} + \lambda \sum\limits_{k = 1}^c {\left\| {{f_k}} \right\|} _H^2$$
$$\mathop {min} \limits_{U,V} {\frac {1}{2} \sum_{n=1}^{N}\sum_{m=1}^{M}{\delta_{nm}}\left|| R_{nm} -U_n^TV_m\right||_F^2+\frac{\lambda_u}{2}\left|| U\right||_F^2+\frac {\lambda_v}{2}\left|| V\right||_F^2}$$
$$\mathop {min} \limits_{U,V} {\frac {1}{2} \sum_{n=1}^{N}\sum_{m=1}^{M}{\delta_{nm}}\left|\left| \left ( \frac {Q_n}{\sqrt{D_{nn}^{row}}} -\frac {P_m}{\sqrt{D_{mm}^{col}}} \right )\right|\right|_F^2+\frac{\eta_q}{2}\left|| Q_n\right||_F^2+\frac {\eta _r}{2}\left|| P_m\right||_F^2}$$

$$min \frac {1}{2}\left |\left| AU -Q \right|\right |_F^2+\frac {\gamma _a}{2}\left |\left |A \right|\right |_F^2$$
$$min \frac {1}{2}\left |\left| BV -P \right|\right |_F^2+\frac {\gamma _a}{2}\left |\left |B \right|\right |_F^2$$
$\mathop {min} \limits_{U,V} O(U,V,Q,P,A,B)= {\frac {1}{2} \sum_{n=1}^{N}\sum_{m=1}^{M}{\delta_{nm}}\left|| R_{nm} -U_n^TV_m\right||_F^2+\frac{\lambda_u}{2}\left|| U\right||_F^2+\frac {\lambda_v}{2}\left|| V\right||_F^2}+{\frac {1}{2} \sum_{n=1}^{N}\sum_{m=1}^{M}{\delta_{nm}}\left|\left| \left ( \frac {Q_n}{\sqrt{D_{nn}^{row}}} -\frac {P_m}{\sqrt{D_{mm}^{col}}} \right )\right|\right|_F^2+\frac{\eta_q}{2}\left|| Q_n\right||_F^2+\frac {\eta _r}{2}\left|| P_m\right||_F^2}+\frac {1}{2}\left |\left| AU -Q \right|\right |_F^2+\frac {\gamma _a}{2}\left |\left |A \right|\right |_F^2+\frac {1}{2}\left |\left| BV -P \right|\right |_F^2+\frac {\gamma _a}{2}\left |\left |B \right|\right |_F^2$

$$\left |\left| BV -P \right|\right |$$
$$\frac {\partial C}{\partial p_{uf}} = -2q_{ik}+ 2\lambda p_{uk}$$
$$\frac {\partial C}{\partial p_{if}} = -2p_{uk}+ 2\lambda q_{ik}$$
$$p_{uf} = p_{uf} +\alpha (q_{ik}-\lambda p_{uk})$$
$$q_{if} = q_{if} + \alpha (p_{uk} - \lambda q_{ik})$$

$$f = \sum_{m=1}^{M} \sum_{n=1}^{N} \left ( R_{mn} - \sum_{k=1}^{K}P_{mk}Q_{kn} \right )^2$$
$$=\sum_{m=1}^{M} \sum_{n=1}^{N} \left (R_{mn}^2 - 2R_{mn}\sum_{k=1}^{K}P_{mk}Q_{kn} +\left (\sum_{k=1}^{K}P_{mk}Q_{kn}\right )^2\right )$$
$$=\sum_{m=1}^{M} \sum_{n=1}^{N} \left (R_{mn}^2 - 2R_{mn}\sum_{k=1}^{K}P_{mk}Q_{kn} +\left (\sum_{k=1}^{K}P_{mk}^2Q_{kn}^2+2 \sum_{i,j=1 ,i\neq j}^{K} {P_{mi}Q_{in}P_{mj}Q_{jn}}\right )\right )$$
$$\frac {\partial f}{\partial P_{mk}} = \sum_{n=1}^{N}\left(-2R_{mn}Q_{kn}+2Q_{kn}^2P_{mk}+2Q_{kn}\sum_{i=1,i \neq k}^{K}{P_{mi}Q_{in}}\right)$$
$$\frac {\partial f}{\partial P_{mk}} = \sum_{n=1}^{N} \left({-2R_{mn}Q_{kn}+2Q_{kn}^2P_{mk}+Q_{kn}\sum_{i=1}^{K}{P_{mi}Q_{in}} -2 Q_{kn}P_{mk}Q_{kn}} \right)$$
$$\frac {\partial f}{\partial P_{mk}} =\sum_{n=1}^{N} \left( -2R_{mn}Q_{kn}+2Q_{kn}\sum_{i=1}^{K}{P_{mi}Q_{in}}\right)$$
$$\frac {\partial f}{\partial Q_{kn}} =\sum_{n=1}^{N} \left( -2R_{mn}P_{mk}+2P_{mk}\sum_{i=1}^{K}{P_{mi}Q_{in}}\right)$$
$$R \in R^{M \times N} , P \in R^{M \times K} , Q \in R^{K \times N}$$
$$\hat R =P Q$$

用P、Q来趋近R

$$\mathop{\Pi}_{i=1}^{K}A_{mn}^i$$
$$\frac {\partial f}{\partial P_{mk}} = \sum_{n=1}^{N} \left({-2R_{mn}Q_{kn}+2Q_{kn}^2P_{mk}+Q_{kn}\sum_{i=1}^{K}{P_{mi}Q_{in}} -2 Q_{kn}P_{mk}Q_{kn}}$$