Notes for "On the mathematical f

作者: jjx323 | 来源:发表于2020-01-30 13:54 被阅读0次

This is a note for the following paper:
F. Cucker, S. Smale, On the mathematical foundations of learning, Bulletin of The American Mathematical Society, 39(1), 1-49, 2001.

Page 6, Remark 2

In addition, $\sigma_{\rho}^{2} = 0$ , the error above specializes to the error mentioned in that discussion, and the regression function $f_{\rho}$ of $\rho$ coincides with $f_{T}$ execpt for a set of measure zero in $X$ .

Note:
For a given $x$ , we have
$y = \left\{\begin{aligned} & 1 \quad x\in T, \\ & 0 \quad x\notin T. \end{aligned}\right.$
For the regression function, we may have
$f_{\rho}(x) = \int_{Y}yd\rho(y|x) = 1\rho(1|x) + 0\rho(0|x),$
where $\rho(1|x) = 1$ for $x\in T$ and $\rho(1|x) = 0$ for $x\notin T$ . Hence, we find that
$f_{\rho}(x) = \left\{\begin{aligned} & 1 \quad x\in T, \\ & 0 \quad x\notin T, \end{aligned}\right.$
which coincides with $f_{T}$ .

Page 10, line 6

Thus, $\mathcal{H}_d$ is a vector space of dimension
$N = \left(\begin{aligned} & n+d \\ & \quad n \end{aligned}\right).$

Note:
Obviously, the conclusion is correct for $n = 1$ . We employ second mathematical induction to illustrate the result. Suppose the result is correct for $n-1$ to $1$ , let us verify the case $n$ . For the additional dimension, we can let $\alpha_n = 0, 1, \cdots, d$ . Then, the number of possible ways should be
$\left(\begin{aligned} & n-1+d \\ & \quad n-1 \end{aligned}\right) + \left(\begin{aligned} & n-1+d - 1 \\ & \qquad n-1 \end{aligned}\right) + \cdots + \left(\begin{aligned} & n-1 \\ & n-1 \end{aligned}\right).$
Written the above formula in a concise manner, we obtain
$\sum_{i=0}^{d}\left(\begin{aligned} & n-1+i \\ & \quad n-1 \end{aligned}\right) = \sum_{i=0}^{d}\left(\begin{aligned} & n-1+i \\ & \quad\quad i \end{aligned}\right) = \left(\begin{aligned} & n + d \\ & \quad d \end{aligned}\right) = \left(\begin{aligned} & n + d \\ & \quad n \end{aligned}\right).$
These calculations verify the desired conclusions.

Page 21, The proof of Proposition 7

Proposition 7 follows from Lemma 8 by applying the same argument used to prove Theorem B from Proposition 3

Note:
Let $\ell = \mathcal{N}\left(\mathcal{H}, \frac{\alpha\epsilon}{4M}\right)$ and consider $f_1, \cdots, f_\ell$ such that the disks $D_j$ centered at $f_j$ and with radius $\frac{\alpha\epsilon}{4M}$ cover $\mathcal{H}$ . Then for every $f\in D_j$ , we have $\|f-f_j\|_{\infty} \leq \frac{\alpha\epsilon}{4M}$ . Employing Lemma 8, we find that
$\begin{align} & \text{Prob}_{z\in Z^m}\left\{ \sup_{f\in D_j} \frac{ \mathcal{E}_{\mathcal{H}}(f)-\mathcal{E}_{\mathcal{H},z}(f)}{\mathcal{E}_{\mathcal{H}}(f)+\epsilon} \geq 3\alpha\right\} \\ \leq & \text{Prob}_{z\in Z^m}\left\{ \frac{ \mathcal{E}_{\mathcal{H}}(f_j)-\mathcal{E}_{\mathcal{H},z}(f_j)}{\mathcal{E}_{\mathcal{H}}(f_j)+\epsilon} \geq 3\alpha\right\} \leq e^{-\frac{\alpha^2 m\epsilon}{8M^2}}. \end{align}$
Proposition 7 has been proved.

Page 27, Proof of Theorem 3

First note that by replacing $A$ by $A^s$ we can reduce the problem in both part (1) and (2) to the case $s=1$

Note:
Since $s > r > 0$ is equivalent to $1 > r' > 0$ with $r' = \frac{r}{s}$ . From the proof, especially the formula of $\hat{t}$ , we know that
$\min_{b\in H}\|b-a\|^2 + \gamma \|A^{-1}b\|^2 \leq \gamma^{r'} \|A^{-r'}a\|^2$
holds true when $0<r'<1$ . Replacing $A$ with $A^{s}$ , we obtain
$\min_{b\in H}\|b-a\|^2 + \gamma \|A^{-s}b\|^2 \leq \gamma^{r'} \|A^{-sr'}a\|^2.$
Finally, we arrive at
$\min_{b\in H}\|b-a\|^2 + \gamma \|A^{-s}b\|^2 \leq \gamma^{r/s} \|A^{-r}a\|^2$
with $0 < r< s$ . Similarly, we can deduce the estimation (2). Here, the result (1) is slightly different from the statment in Theorem 3. It may be a small mistake.

Notes for "On the mathematical f

Page 6, Remark 2

Page 10, line 6

Page 21, The proof of Proposition 7

Page 27, Proof of Theorem 3

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