Intereting Posts

Determining the Asymptotic Order of Growth of the Generalized Harmonic Function?
If $x \equiv 1 \pmod 3$ and $x \equiv 0 \pmod 2$, what is $x \pmod 6$?
How big can a separable Hausdorff space be?
Limit of the infinite sum of $\frac{n}{2^n}$?
The Dido problem with an arclength constraint
Evaluating $\int \frac{1}{{x^4+1}} dx$
Distribution theory book
How can I express $\sum_{k=0}^n\binom{-1/2}{k}(-1)^k\binom{-1/2}{n-k}$ without using summations or minus signs?
Calculation of the moments using Hypergeometric distribution
Why is completeness theorem true?
Another Laplace transform of a function with square roots.
Has error correction been “solved”?
Proving the inverse of a relation exists
Find all positive integers $n$ such that $\phi(n)=6$.
Proving theorem connecting the inverse of a holomorphic function to a contour integral of the function.

In my machine learning course I’m asked to proof that

$MSE(w) = \frac{1}{N} \sum_{n=1}^{N}[y_n – f(x_n)]^2$

when $f$ is of the form $w_0 + x_{n1} w_1 + \cdots + x_{nD} w_D$ is a convex function on the parameters $w_1,\cdots, w_D$.

- If the Minkowski sum of two convex closed sets is a Euclidean ball, then can the two sets be anything other than Euclidean balls?
- Characterization convex function.
- Differentiable Strictly Convex Function on Interval
- Minimizing $\sum_{i=1}^n \max(|x_i - x|, |y_i - y|)$ - Sum of Max of Absolute Values
- A convex function is differentiable at all but countably many points
- Legendre transform of a norm

I’m a bit confused on how to do it though. My approach goes as follows:

I first proof that the sum of two convex functions is convex. Then I try to develop the value of $(y_n – f(x_n))^2$ however I arrive to the point where I have to proof that a function of type $w_iw_j$ is convex which is not true as shown in this post.

How can I proceed?

- How do you prove that $\{ Ax \mid x \geq 0 \}$ is closed?
- Example of a function such that $\varphi\left(\frac{x+y}{2}\right)\leq \frac{\varphi(x)+\varphi(y)}{2}$ but $\varphi$ is not convex
- Convex hull of orthogonal matrices
- Are there necessary and sufficient conditions for Krein-Milman type conclusions?
- How to prove that $e^x$ is convex?
- Are there necessary and sufficient conditions so that every element in a partially ordered set is either the least element or in the upset of an atom?
- Show that $f$ is convex if and only if $f\left( \sum_{i=1}^m\lambda_ix_i \right) \leq \sum_{i=1}^m\lambda_if(x_i)$
- If $f$ is midpoint convex, continuous, and two times differentiable, then for any $a, b \in \mathbb{R}$, there exists $c$ such that $f''(c) \geq 0$
- Every closed convex cone in $ \mathbb{R}^2 $ is polyhedral
- Proof of Non-Convexity

Hint: It may be easier if you write the MSE in matrix-vector form as

$$

\operatorname*{MSE}(w) = \frac{1}{N}\|y – Xw\|^2.

$$

From here you can show that

$$

\operatorname*{MSE}(w) + \nabla\operatorname*{MSE}(w)\cdot(z – w) \leq \operatorname*{MSE}(z)

$$

for all $w,z\in\mathbb{R}^{D+1}$.

- When do free variables occur? Why allow them? What is the intuition behind them?
- $\sum_{m=1}^{\infty}{\frac{e^{-a m^2}}{m^2}}$ as an Integral
- Substituting for Taylor series
- Bounded inverse operator
- Integrate $I(a) = \int_0^{\pi/2} \frac{dx}{1-a\sin x}$
- If a rational number has a finite decimal representation, then it has a finite representation in base $b$ for any $b>1?$
- How many vertices do you need so that all squares are not directly connected?
- Canonical symplectic form on cotangent bundle of complex manifold
- a question about germs of functions
- (co)reflector to the forgetful functor $U:\mathbf{CMon} \to \mathbf{ Mon}$
- Finding a generating function of a series
- Finding an upperbound for $\sum_{i=2}^{n}\bigg(\prod_{k=2}^{i}\dfrac{p_k-2}{p_k}\bigg)$
- Alternative definition of hyperbolic cosine without relying on exponential function
- Trace of the $n$-th symmetric power of a linear map
- The torsion submodule of $\prod \mathbb{Z}_p$ is not a direct summand of $\prod \mathbb{Z}_p$