I am using a loss function in quadratic form: $$ l(w) = w^TAw \\ s.t. w^TBw=1 $$ where both $A$ and $B$ are symmetric positive definite. $A$ and $B$ are between-class scatter matrix and within-class scatter matrix, respectively. An explanation of between-class/within-class scatter matrix can be found here. I want to estimate $w$ by minimizing $l(w)$. If there are outliers in my dataset, they could have great (or even adversarial) impact on my estimation of $w$, through $l(w)$. I am wondering if there is any techniques to robustify $l(w)$. For example, we use Huber loss instead of square error loss to reduce the impact of outliers. Any paper/tutorial/comments are appreciated!
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1$w=0$ minimizes that function for all data, but this probably isn't what you're after. Can you explain more about what problem you're trying to solve and what $w$ is? – Sycorax Nov 15 '16 at 20:15
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@Sycorax Sorry I forgot to put down a constraint. I have added it now. Thanks – MarsPlus Nov 15 '16 at 20:45
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Thank you--adding constraints helps. Could you also indicate how your dataset might be connected to the quantities $w$, $A$, and/or $B$? – whuber Nov 15 '16 at 20:49
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@whuber the data is connected to the loss by between-class scatter matrix $A$ and within-class scatter matrix $B$, as in Fisher discriminant analysis. Thanks! – MarsPlus Nov 15 '16 at 21:03