## An intuition to an tangent plane to a surface

To me: Page 126 of Cracking the GRE Math Subject Test. A rigorous proof can be found here (Calculus, Howard Anton). A note of stress should be on “any smooth curve C on the surface”. Below is my attempt interpretation before looking at any proofs. While it cannot be considered as a proof, it shows [...]

## Proof of lower-rank matrix factorization

Let and and  , prove that . We can determine  by applying row operations on . First, we will let the second row of  after the first row operation be: . Next, the third row of  after the second row operation would be: . Soon, we will come to () which is a linear combination [...]

## Entropy 2016, bảng A

It took me quite a lot of time to understand the question, though. I do not post the test here as I am not sure if I am permitted to do that. This post serves for my personal use. Even though the original questions are written in Vietnamese, I will be writing my solutions in [...]

## Triplet Loss’ derivative of VGG Face

We starts with the formula (1) of the paper.     We have:             By chain rule, we have:     We also have: – the th element of .   – the th element of . So:

## [Notes] Learning From Data – A Short Course: e-Chapter 9

Page 30: Why ? My understanding:   as the events and are independent, the same goes for the event . Note that both data points and may exist in dataset so if is a deterministic hypothesis then it is obviously  as . If is a non-deterministic hypothesis,  is still independent from the random target function (even “ are [...]

## Learning From Data – A Short Course: Exercise 9.6

Try to build some intuition for what the rotation is doing by using the illustrations in Figure 9.1 to qualitatively answer these questions. (a) If there is a large offset (or bias) in both measured variables, how will this affect the ‘natural axes’, the ones to which the data will be rotated? Should you perform [...]

## Learning From Data – A Short Course: Exercise 9.4

Let and  be independent with zero mean and unit variance. You measure inputs and . (a) What are variance (), variance () and covariance ? First, we have:                 Now, we consider and get: Expected values:             Variance:           [...]

## Learning From Data – A Short Course: Exercise 8.15

Consider two finite-dimensional feature transforms and and their corresponding kernels and . (a) Define . Express the corresponding kernel of in terms of and .     (b) Consider the matrix and let be the vector representation of the matrix (say, by concatenating all the rows). Express the corresponding kernel of in terms of and [...]