# Calculating gradient for inner product of variable matrix

The problem is formulated as follows:

Let $$U\in\mathbb{R}^{d\times R}$$ denote the variable matrix. $$A_i\in\mathbb{R}^{d\times d}$$ and $$y_i$$ is a scalar $$f(U) = \frac{1}{2m}\sum^m_{i=1}(\langle{A_i, UU^T}\rangle-y_i)^2$$

Compute the gradient of $$f(U)$$ over $$U$$.

I have never dealt with inner products much before and am confused how to take the gradient for this function. I know that

$$\frac{d}{dt} \langle f, g \rangle = \langle f(t), g^{\prime}(t) \rangle + \langle f^{\prime}(t), g(t) \rangle$$

But if you do this then the matrix dimensions do not work. Also not sure if this is even correct as you are dealing with a variable matrix. Any help would be great as I am pretty lost on how to begin.

• I'm confused by the quantity $\Big(\langle A_i,UU^T\rangle-y_i\Big)$ -- the first term is a scalar but $y_i$ is a matrix. Either there's a missing identity matrix multiplying the first term, or $y_i$ is actually a scalar. Is $f(U)$ itself a scalar or a matrix? – greg Oct 17 at 17:42
• Your right, it is a scalar, editing it. I was trying to write it concisely and got mixed up. $f(U)$ is a scalar – mkohler Oct 17 at 19:17
• Which is the inner product? I guess it is $<A,B>=\sum_{ij} A_i B_i$ ? – Miguel Oct 17 at 19:26
• @Miguel You are correct – mkohler Oct 17 at 19:58

For ease of typing, I'll use a colon to denote the inner product, i.e. $$A:B = \langle A,B\rangle$$ Define a vector $$v$$, whose $$i^{th}$$ component is given by $$v_i = A_i:(UU^T) - y_i$$ and a matrix $$M$$ equal to the mean of the $$A_i$$ matrices (weighted by the components of $$v$$) $$M = \frac 1m\sum_{i=1}^m v_iA_i$$ Then the function can be written as $$f = \frac 1{2m}\;\sum_{i=1}^m v_i\,v_i$$ Calculate the gradient of this function as follows \eqalign{ df &= \frac 1m\;\sum_{i=1}^m v_i\,dv_i \\ &= \frac 1m\;\sum_{i=1}^m v_i\,A_i:d(UU^T) \\ &= M:d(UU^T) \\ &= M:(dU\,U^T+U\,dU^T) \\ &= (M+M^T):(dU\,U^T) \\ &= \left(M+M^T\right)U:dU \\ \frac{\partial f}{\partial U} &= \left(M+M^T\right)U \\ }
• An alternative expression for the inner product is $A:B={\rm Tr}(A^TB)\;$ and the step that you're asking about follows from well-known properties of the trace. – greg Oct 18 at 17:38