Hello pricey customer to our community We will proffer you an answer to this query matrix evaluation – Gradient Descent for Markov Dynamics ,and the respond will breathe typical via documented data sources, We welcome you and proffer you fresh questions and solutions, Many customer are questioning in regards to the respond to this query.

matrix evaluation – Gradient Descent for Markov Dynamics

Note that $$A$$ solely seems within the mixture $$M=A-BC$$, so the by-product with respect to $$A$$ equals the by-product with respect to $$M$$; The duty $$f(A)$$ is given by
$$f(A)=||(A – BC)^Nv – w||_2^2=(M^Nv)^T M^Nv+w^Tw-2w^TM^Nv.$$
For a unostentatious illustration, let me first deem a scalar perturbation, $$f(A+epsilon I)=f(A)+epsilon df/dA$$, with by-product
$$frac{df}{dA} =2N(M^Nv-w)^TM^{N-1}v.$$

Next deem the by-product with respect to a given matrix component $$A_{ij}$$ of $$A$$. The expressions are extra prolonged, mainly every matrix $$M$$ offers a several time period so we’ve a sum $$sum_{okay=1}^N$$ as a substitute of the issue $$N$$:
$$frac{partial f}{partial A_{ij}}= 2sum_{okay=1}^Nsum_{p,q=1}^n (M^Nv-w)_q (M^{k-1})_{qi} (M^{N-k})_{jp} v_p.$$
Note that the scalar perturbation is the vestige of the matrix of elementwise perturbations, $$df/dA=sum_{i=1}^n partial f/partial A_{ii}$$.

As celebrated within the observation, my outcome for the elementwise by-product doesn’t appear to conform with the one from a web-based calculator. Let me bridle a unostentatious illustration, $$N=2$$, $$w=0$$, $$v_p=delta_{p1}$$. Then
$$f=sum_{p,q,r}M_{pq}M_{q1}M_{pr}M_{r1}.$$
Direct analysis of the by-product with respect to $$M_{22}$$ offers
$$frac{partial f}{partial M_{22}}=2M_{21}(M^2)_{21},$$
in settlement with the common method above. The on-line calculator would give 0.

we are going to proffer you the answer to matrix evaluation – Gradient Descent for Markov Dynamics query by way of our community which brings all of the solutions from a number of dependable sources.