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After Algorithm 1 returns, we get the two factor matrices W and H . These two matrices can be used to compute missing association scores between the row and the column entities. It can also provide prediction score for an association between a known row entity with an new column entity, or a known column entity with a new row entity, or both new row and column entities.

\left\|A - XWH^{T}Y^{T}\left\|{~}^{2}_{F} = \sum_{i=1}^{M}\right\|A_{i,:}-X_{i,:}WH^{T}Y^{T}\right\|{~}_{2}^{2}
\left\|A - XWH^{T}Y^{T}\right\|{~}_{2,1} = \sum_{i=1}^{M}\sqrt{\sum_{j=1}^{N}\left(A-XWH^{T}Y^{T}\right)^{2}_{{ij}}}
\begin{array}{*{20}l} \underset{W,H}{\text{min}} \quad\varphi = \left\|A-XWH^{T}Y^{T}\right\|{~}_{2,1} + \lambda_{1} R(W) + \lambda_{2} R(H)\\ \text{such that},\quad W\geq 0, H\geq 0 \end{array}

Here, we have several options as the regularization function R (·); such as: R_{1}(B) = ||B||^{2}_{F} , R_{2}(B) = \sum _{i=1}^{M}||B_{i,:}||_{1} , R_{3}(B) = \sum _{i=1}^{M}||B_{i,:}||^{0}_{2} and R_{4}(B) = \sum _{i=1}^{M}||B_{i,:}||_{2} . Here, R 1 (·) is the ridge regularization and is adapted in the standard IMC formulation, R 2 (·) is the LASSO regularization which is a non-convex function and difficult to optimize. R 3 (·) involves the 0 norm and is the most desirable [ 11 ], and R 4 (·) employs the 2,1 norm. R 4 (·) was chosen because the function is convex and we can easily optimize the objective function involving this kind of regulizer [ 12 ].

\begin{array}{*{20}l} \underset{W,H}{\text{min}} \quad\varphi = \left\|A-XWH^{T}Y^{T}\right\|{~}_{2,1} + \lambda_{1} \|W\|{~}_{2,1} + \lambda_{2} \|H\|{~}_{2,1}\\ \text{such that},\quad W\geq 0, H\geq 0 \end{array}

In order to solve Eq. 7 , Algorithm 2 can be adapted [ 10 ].

Theorem 1

At convergence, the converged solution of the updating rule in Algorithm satisfies the KKT condition.

\frac{\partial \varphi(W)}{\partial W_{{ik}}}W_{{ik}} = 0, \forall i,k
{}\begin{aligned} \frac{\partial \varphi(W)}{\partial W_{{ik}}}=\\ \sum_{\alpha=1}^{M}\frac{1}{\sqrt{\sum_{\beta=1}^{N}\left(A-XWH^{T}Y^{T}\right)^{2}_{\alpha\beta}}} \cdot\sum_{\beta^{\prime}=1}^{N}\left(A-XWH^{T}Y^{T}\right)_{\alpha\beta^{\prime}}\cdot\\ \frac{\partial}{\partial W_{{ik}}} \left(A-XWH^{T}Y^{T}\right)_{\alpha\beta^{\prime}}+ \lambda_{1}\sum_{\alpha=1}^{m}\frac{1}{\sum_{\beta=1}^{r} W^{2}_{\alpha\beta}}\cdot W_{\alpha\beta}\cdot \frac{\partial W_{\alpha\beta} }{\partial W_{{ik}}}\\ =\sum_{\alpha=1}^{M} D_{\alpha\alpha}\sum_{\beta^{\prime}=1}^{N}\left(A-XWH^{T}Y^{T}\right)_{\alpha\beta^{\prime}}\left(-X^{T}e_{M}e^{T}_{N}YH\right)_{{ik}} \\ + \lambda_{1}P_{{ii}}W_{{ik}} \end{aligned}
D_{{ii}} = 1 \bigg/\sqrt{\sum_{j=1}^{N} \left(A-XWH^{T}Y^{T}\right)^{2}_{{ij}}}
P_{{ii}} = 1 \bigg/\sqrt{\sum_{j=1}^{r} W^{2}_{{ij}}}
{}\begin{aligned} \frac{\partial \varphi(W)}{\partial W_{{ik}}}=-\left(e_{m}e^{T}_{M}Ae_{N}e^{T}_{M}De_{M}e^{T}_{m}X^{T}e_{M}e^{T}_{N}YH\right)_{{ik}}\\ \quad \left(\!e_{m}e^{T}_{M}XWH^{T}Y^{T}e_{N}e^{T}_{M}De_{M}e^{T}_{m}X^{T}e_{M}e^{T}_{N}YH\!\right)_{{ik}}\\ \quad+ \lambda_{1}P_{{ii}}W_{{ik}} \end{aligned}
\begin{array}{@{}rcl@{}}{} \left[-\left(e_{m}e^{T}_{M}Ae_{N}e^{T}_{M}De_{M}e^{T}_{m}X^{T}e_{M}e^{T}_{N}YH\right)_{{ik}}\right.\\ {} \quad\!\!\! + \left(e_{m}e^{T}_{M}XWH^{T}Y^{T}e_{N}e^{T}_{M}De_{M}e^{T}_{m}X^{T}e_{M}e^{T}_{N}YH\right)_{{ik}} \qquad\\ \qquad\qquad \qquad\qquad\qquad\left.{\vphantom{a^{a^{a}}}} + \lambda_{1}P_{{ii}}W_{{ik}}\right]W_{{ik}} =0, \forall i,k \end{array}
\begin{array}{@{}rcl@{}} \left[-\left(e_{m}e^{T}_{M}Ae_{N}e^{T}_{M}De_{M}e^{T}_{m}X^{T}e_{M}e^{T}_{N}YH\right)_{{ik}}\right.\\ \quad\!\!\! + \left(e_{m}e^{T}_{M}XW^{*}H^{T}Y^{T}e_{N}e^{T}_{M}De_{M}e^{T}_{m}X^{T}e_{M}e^{T}_{N}YH\right)_{{ik}}\\ \qquad\qquad\qquad\qquad\qquad\qquad \left.{\vphantom{^{a^{a}}}}+ \lambda_{1}P_{{ii}}W^{*}_{{ik}}\right]W^{*}_{{ik}} =0 \end{array}

This is identical to Eq. CORRAL Womens Contrast Side Embroidery Cowgirl Boot Snip Toe A3170 Black vosRO6GCd6
. Thus, the converged solution W satisfies the KKT condition. □

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Dan Crawford

I love the view from his house, but aye look at Klay living his best life #KlayThompson

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Here’s your weekly summer Klay Thompson update. It looks like he made back to the states after dominating China for the second year in a row just in time to celebrate our country’s birthday yesterday. Just a little backyard BBQ at some Hollywood Hills house featuring somewhere between 25-50 women (I won’t call them IG models, but we can assume) all chilling around the hot tub and pool while Klay practices his chipping stroke. China Klay is officially America Klay once again.

China Klay Thompson Dominates School Beauty Pageant

Klay Thompson Involved In Pick-Up Game Scuffle In China

When people say “living your best life” this is what they mean. You’re still on the celebration tour of another championship, you just did your thing in China for a few weeks, now it’s time to do it up in back home American style. Food, drinks, sun, pool, and lots of ladies. I can’t imagine many people are living harder than Klay right now.

IN OTHER NEWS: Serena Williams Wins Her First Wimbledon Match as a Mom
Serena Williams Wins Her...

I would say Warriors fans held their breath here, but if we’re being honest Klay could fall off the face of the Earth today and Golden State would still be the betting favorite in Vegas.

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A post shared by Klay Thompson Fanpage! (@klayfor3) on

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Jul 5, 2018

Dan Crawford

Berkshire Conference of Women Historians

Perhaps you already know us for our triennial meeting, The Berkshire Conference on the History of Women ("The Big Berks") Launched in 1973, it is now the largest gathering of its kind in the world. Our 2017 meeting will be at Hofstra University, less than an hour's drive outside New York City.

Members are invited to our annual retreats, where many scholars begin their relationship to the organization in an atmosphere designed to promote friendship, relaxation, and networking. Feel free to click around our site to read more about our history, our governance structure, our prizes and our committees. We hope that when you learn more about us you will want to join at the level that is comfortable for you.

Our Vision Fostering friendship, and the exchange of ideas, among a global network of feminist historians. Our Mission Promoting the interests of women historians, their allies and friends. OurPurpose The Berkshire Conference of Women … Aisun Womens Classic Warm Comfy Thicken Flat Ankle Snow Boots Black NxCxA

The Berkshire Conference of Women Historians was founded in 1930 by Louise Fargo Brown of Vassar College and Louise Ropes Loomis of Wells College.Initially called the Lakeville History Group, after the Connecticut retreat where early meetings were … Read More »

Too little is still known about the history of women's history and the role women have played more generally in the historical profession. The Berkshire Conference of Women Historians past, and the archives for the triennial conferences, are kept at … Read More »

The leadership of the Berks is guided by its officers. Questions about the triennial conference, initiatives you would like the Berkshire Conference to undertake, or service on Berkshire Conference Committees should go to the President. Questions … Read More »

The 2020 Berkshire Conference on the History of Women, Genders and Sexualities will be held in Baltimore, Maryland. The exact dates will be announced shortly. Please stay tuned for details. Keep up-to-date with Big Berks News Sign me up to receive information about the … Read More »


Become a Member The Berkshire Conference of Women Historians is a vital network of scholars that welcomes all women in the historical profession. We offer two kinds of events: our triennial … Belleville Mens Hot Weather Tan Tactical Boots 310 Tan mf1Vp6ZWK

Fall meeting of the Berkshire Conference of Women's Historians June 1-3, 2018 The Berkshire Conference of Women Historians invites you to attend its annual business meeting, better known as the Little Berks from Friday-Sunday, June 1-3, 2018 at Pheasant Run Resort in Mt. Pleasant, Illinois, a … Read More »

The Berkshire Conference of Women Historians awards two annual book and article prizes in the following categories: An article in the fields of the history of women, gender, and/or sexuality. An article in any field of history other than the history of women, gender, and/or sexuality. A first book that deals … Read More »

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