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Question:

Consider a series of functions $f_1,\cdots,f_d:\mathbb{R}^d\to\mathbb{R}$ such that \begin{align}f_1(v_1,\cdots,v_d)&=0\\ &\vdots\\ f_d(v_1,\cdots,v_d)&=0\end{align}

for $v_1,v_2,\cdots,v_d\in \mathbb{R}$

Fix $\textbf{x},\textbf{y}\in \mathbb{R}^d$ and let $$g(\lambda)=f_i(\textbf{x}+\lambda(\textbf{y}-\textbf{x}))$$ for $\lambda \in \mathbb{R}$

Write down a Taylor series for $g$ near $\lambda=0$ to second order.

I know that this Taylor expansion will have the form $$g(\lambda)=g(0)+\lambda g'(0)+\frac{\lambda ^2}{2}g''(0)$$ (differentiating wrt $\lambda$) however I am struggling to actually compute this.

I have computed the following:

\begin{align}g'(\lambda)&=(\textbf{y}-\textbf{x}) f_i '(\textbf{x}+\lambda (\textbf{y}-\textbf{x}))\\ g''(\lambda)&=(\textbf{y}-\textbf{x})^2 f_i ''(\textbf{x}+\lambda (\textbf{y}-\textbf{x}))\end{align}

however $g''(\lambda)$ involves the square of a vector, which I feel is incorrect.

Can someone either point me in the right direction or explain to me where I am going wrong please.


Edit: I found this related question and have come up with a potential solution. Can someone verify it for me please?

$$g(\lambda) = f_i(\textbf{x}) + \lambda \sum_{j=1}^d \frac{\partial f_i(\textbf{x})}{\partial x_j}(y_j-x_j) + \frac{\lambda^2}2 \sum_{j,k=1}^d\frac{\partial f_i(\textbf{x})}{\partial x_j\partial x_k}(y_j-x_j)(y_k-x_k)$$

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  • $\begingroup$ Why do you need $v_1,v_2,\ldots,v_d$? $\endgroup$ – Batominovski Nov 28 '17 at 16:52
  • $\begingroup$ @Batominovski They might not be required for this part of the question - this is part of a longer series of questions but I'm struggling to get started. I merely included them in case they were useful here $\endgroup$ – lioness99a Nov 28 '17 at 16:53
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    $\begingroup$ OK. I thought you mistyped something. $\endgroup$ – Batominovski Nov 28 '17 at 16:56
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These expansions get confusing quickly with vector valued variables and functions as arguments, so I'll renotate a bit for (hopefully) more clarity and derive the result using just the multivariate chain rule.

Let $\lambda\in\mathbb{R}$, $f_i : \mathbb{R}^n \rightarrow\mathbb{R}$, $u : \mathbb{R} \rightarrow\mathbb{R}^n$, $x,y\in\mathbb{R}^n$, and $$ g(\lambda)=f_i(x+\lambda (y-x)) = f_i(u(\lambda))$$ where $u(\lambda)=x+\lambda(y-x)$. Notice that $\partial_\lambda u = y-x$. Then the second-order univariate Taylor expansion about $\lambda=a$ is $$ T_g(\lambda|a) = g(a)+[\lambda - a]\partial_\lambda g(a) + \frac{1}{2}\partial_{\lambda\lambda} g(a) [\lambda - a]^2 $$ where $\partial_\lambda=\partial/\partial \lambda$. We really just need $\partial_{\lambda} g(a)$ and $\partial_{\lambda\lambda} g(a)$. The first one, using the multivariate chain rule: \begin{align} \partial_\lambda g(\lambda) &= \sum_k \left.\frac{\partial f_i}{\partial u_k} \right\rvert_{u(\lambda)} \partial_\lambda u_k(\lambda) \\ &= \nabla f_i(u(\lambda))^T [y-x] \tag{1} \end{align} Now from (1) we can evaluate at the special case of $\lambda=0$: \begin{align} \partial_\lambda g(0) &= \sum_k \left.\frac{\partial f_i}{\partial u_k} \right\rvert_{u(0)} \partial_\lambda u_k(0) = \sum_k \left. \frac{\partial f_i}{\partial x_k}\right\rvert_{x} [y_k - x_k] \\ &= \sum_k [y_k - x_k] \frac{\partial}{\partial x_k} f_i(x) = (y-x)^T \nabla_x f_i(x) \tag{2} \end{align} because $$ \left.\frac{\partial u_k}{\partial x_k}\right|_{\lambda=0} = \left.(1-\lambda)\right|_{\lambda=0} =1 $$ so $$ \left.\frac{\partial f_i}{\partial x_k}\right|_{\lambda=0} = \left.\frac{\partial f_i}{\partial u_k}\right|_{u(0)} \left. \frac{\partial u_k}{\partial x_k}\right|_{\lambda = 0} \implies \left.\frac{\partial f_i}{\partial u_k}\right|_{u(0)} = \left.\frac{\partial f_i}{\partial x_k}\right|_{x} $$ Ok, now for the second derivative. First, I want to define $q :\mathbb{R}^n \rightarrow \mathbb{R}^n$ as $q(u(\lambda)) = \nabla f_i(u(\lambda))$, meaning that $q_j : \mathbb{R}^n \rightarrow \mathbb{R}$ can be written $q_j(u(\lambda)) = (\partial f_i/\partial u_j)|_{u(\lambda)}$. This means that the Hessian matrix $\mathcal{H}$ of $f$ has rows given by $$\nabla q_j(u(\lambda))^T = \left. \left[ \frac{\partial^2 f_i}{\partial u_1 \partial u_j},\cdots,\frac{\partial^2 f_i}{\partial u_n \partial u_j} \right]\right\rvert_{u(\lambda)}, $$ that is, $ \mathcal{H}[f_i(u(\lambda))] $ has components given by: $$ \mathcal{H}_{\ell k} = \left.\frac{\partial^2 f_i}{\partial u_\ell \partial u_k}\right|_{u(\lambda)} = \nabla q_\ell(u(\lambda))_k $$ Anyway, we can then compute (using the chain rule again): $$ \frac{\partial}{\partial \lambda} q_j(u(\lambda)) = \nabla q_j(u(\lambda))^T \frac{\partial u}{\partial \lambda} = \nabla q_j(u(\lambda))^T[y-x] $$ Alright, so for the second derivative, start with the result from (1): \begin{align} \partial_{\lambda\lambda} g(\lambda) &= \partial_\lambda \left[ \nabla f_i(u(\lambda))^T [y-x] \right] \\ &= \sum_j (y_j - x_j)\frac{\partial}{\partial\lambda}\frac{\partial}{\partial u_j}f_i(u(\lambda)) \\ &= \sum_j (y_j - x_j)\frac{\partial}{\partial\lambda}q_j(u(\lambda)) \\ &= \sum_j (y_j - x_j)\nabla q_j(u(\lambda))^T[y-x] \\ &= \sum_j (y_j - x_j) \sum_k \nabla q_j(u(\lambda))_k[y_k-x_k] \\ &= \sum_j (y_j - x_j) \sum_k \frac{\partial^2 f_i(u(\lambda))}{\partial u_j \partial u_k} [y_k-x_k] \\ &= (y-x)^T \mathcal{H}[f_i(u(\lambda))](y-x) \tag{3} \end{align} Next, let's use the following relation, which will help us get rid of $u$: \begin{align} \frac{\partial^2 f_i}{\partial x_\ell \partial x_k} &= \frac{\partial}{\partial x_\ell} \frac{\partial f_i}{\partial x_k} = \frac{\partial}{\partial x_\ell} \left(\frac{\partial f_i}{\partial u_k}\frac{\partial u_k}{\partial x_k}\right) = \underbrace{\frac{\partial u_k}{\partial x_k}}_{1-\lambda} \; \underbrace{ \frac{\partial^2 f_i}{\partial x_\ell \partial u_k} }_{\partial q_k(u(\lambda))/ \partial {x_\ell}} + \frac{\partial f_i}{\partial u_k}\underbrace{\frac{\partial}{\partial x_\ell}\frac{\partial u_k}{\partial x_k}}_0 \\ &= (1-\lambda)\frac{\partial}{\partial u_k} \frac{\partial f_i}{\partial{x_\ell}} = (1-\lambda)\frac{\partial}{\partial u_k}\left[ \frac{\partial f_i}{\partial{u_\ell}}\frac{\partial u_\ell}{\partial{x_\ell}} \right] = (1-\lambda)^2\frac{\partial^2 f_i(u(\lambda))}{\partial u_j \partial u_k} \end{align} So, at $\lambda = 0$ we have $$ \left.\frac{\partial^2 f_i(u(\lambda))}{\partial u_j \partial u_k}\right|_{u(0)} = \frac{\partial^2 f_i(x)}{\partial x_\ell \partial x_k} \tag{4} $$ which means that (using (3) and (4)) $$ \partial_{\lambda\lambda} g(0) = \sum_{j,k} (y_j - x_j) \frac{\partial^2 f_i(u(0))}{\partial u_j \partial u_k} [y_k-x_k] = \sum_{j,k} [y_j - x_j] \frac{\partial^2 f_i(x)}{\partial x_\ell \partial x_k} [y_k-x_k] \tag{5} $$ Altogether, using (1) and (3), the general form of the second order Taylor expansion of $g(\lambda)$ expanded about $a$ can be written as $$ T_g(\lambda|a) = g(a) + [\lambda - a]\nabla f_i(u(a))^T [y-x] + \frac{1}{2}[\lambda - a]^2 (y-x)^T \mathcal{H}[f_i(u(a))](y-x) $$ and in the desired special case where $a=0$ we get (using (2) and (5)): \begin{align} T_g(\lambda|0) &= f_i(x) + \lambda (y-x)^T \nabla_x f_i(x) + \frac{\lambda^2}{2} (y-x)^T \mathcal{H}[f_i(x)](y-x) \\ &= f_i(x) + \lambda \sum_\ell (y_\ell - x_\ell)\partial_x f_i(x) + \frac{\lambda^2}{2} \sum_{j,k}(y_j-x_j)(y_k - x_k) \frac{\partial^2 f_i(x)}{\partial x_j \partial x_k} \end{align} exactly as you did. :D

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