The main contribution of [Understanding the difficulty of training deep feedforward neural networks](http://jmlr.org/proceedings/papers/v9/glorot10a/glorot10a.pdf) by Glorot et al. is a **normalized weight initialization**

$$W \sim U \left [  - \frac{\sqrt{6}}{\sqrt{n_j + n_{j+1}}}, \frac{\sqrt{6}}{\sqrt{n_j + n_{j+1}}} \right ]$$

where $n_j \in \mathbb{N}^+$ is the number of neurons in the layer $j$.

Showing some ways **how to debug neural networks** might be another reason to read the paper.

The paper analyzed standard multilayer perceptrons (MLPs) on a artificial dataset of $32 \text{px} \times 32 \text{px}$ images with either one or two of the 3 shapes: triangle, parallelogram and ellipse. The MLPs varied in the activation function which was used (either sigmoid, tanh or softsign).

However, no regularization was used and many mini-batch epochs were learned. It might be that batch normalization / dropout might change the influence of initialization very much.

Questions that remain open for me:

* [How is weight initialization done today?](https://www.reddit.com/r/MLQuestions/comments/4jsge9)
* Figure 4: Why is this plot not simply completely dependent on the data?
* Is softsign still used? Why not?
* If the only advantage of softsign is that is has the plateau later, why doesn't anybody use $\frac{1}{1+e^{-0.1 \cdot x}}$ or something similar instead of the standard sigmoid activation function?

This paper deals with the question what / how exactly CNNs learn, considering the fact that they usually have more trainable parameters than data points on which they are trained.

When the authors write "deep neural networks", they are talking about Inception V3, AlexNet and MLPs.

## Key contributions

* Deep neural networks easily fit random labels (achieving a training error of 0 and a test error which is just randomly guessing labels as expected). $\Rightarrow$Those architectures can simply brute-force memorize the training data.
* Deep neural networks fit random images (e.g. Gaussian noise) with 0 training error. The authors conclude that VC-dimension / Rademacher complexity, and uniform stability are bad explanations for generalization capabilities of neural networks
* The authors give a construction for a 2-layer network with $p = 2n+d$ parameters - where $n$ is the number of samples and $d$ is the dimension of each sample - which can easily fit any labeling. (Finite sample expressivity). See section 4.


## What I learned

* Any measure $m$ of the generalization capability of classifiers $H$ should take the percentage of corrupted labels ($p_c \in [0, 1]$, where $p_c =0$ is a perfect labeling and $p_c=1$ is totally random) into account: If $p_c = 1$, then $m()$ should be 0, too, as it is impossible to learn something meaningful with totally random labels.
* We seem to have built models which work well on image data in general, but not "natural" / meaningful images as we thought.


## Funny

> deep neural nets remain mysterious for many reasons

> Note that this is not exactly simple as the kernel matrix requires 30GB to store in memory. Nonetheless, this system can be solved in under 3 minutes in on a commodity workstation with 24 cores and 256 GB of RAM with a conventional LAPACK call.


## See also

* [Deep Nets Don't Learn Via Memorization](https://openreview.net/pdf?id=rJv6ZgHYg)

How can we learn causal relationships that explain data? We can learn from non-stationary distributions. If we experiment with different factorizations of relationships between variables we can observe which ones provide better sample complexity when adapting to distributional shift and therefore are likely to be causal.

If we consider the variables A and B we can factor them in two ways:

$P(A,B) = P(A)P(B|A)$ representing a causal graph like $A\rightarrow B$

$P(A,B) = P(A|B)P(B)$ representing a causal graph like $A \leftarrow B$

The idea is if we train a model with one of these structures; when adapting to a new shifted distribution of data it will take longer to adapt if the model does not have the correct inductive bias. For example let's say that the true relationship is $A$=Raining causes $B$=Open Umbrella (and not vice-versa). Changing the marginal probability of Raining (say because the weather changed) does not change the mechanism that relates $A$ and $B$ (captured by $P(B|A)$), but will have an impact on the marginal $P(B)$. 

So after this distributional shift the function that modeled $P(B|A)$ will not need to change because the relationship is the same.  Only the function that modeled $P(A)$ will need to change. Under the incorrect factorization $P(B)P(A|B)$, adaptation to the change will be slow because both $P(B)$ and $P(A|B)$ need to be modified to account for the change in $P(A)$ (due to Bayes rule).

Here a difference in sample complexity can be observed when modeling the joint of the shifted distribution.  $B\rightarrow A$ takes longer to adapt:
https://i.imgur.com/B9FEmA7.png

Here the idea is that sample complexity when adapting to a new distribution of data is a heuristic to inform us which causal graph inductive bias is correct.

Experimentally this works and they also observe that when models have more capacity it seems that the difference between the models grows.

This summary was written with the help of Yoshua Bengio.

This paper presents a method to extract motion (dynamic) and skeleton / camera-view (static) representations from the video of a person represented as a 2D joints skeleton. This decomposition allows transferring the motion to different skeletons (retargeting) and many more.  It does so by utilizing deep neural networks. 
https://i.imgur.com/J5jBzcs.png

The architecture consists of motion and skeleton / camera-view encoders that decompose an input sequence of 2D joint positions into latent spaces and a decoder that reconstructs a sequence from such components. The motion vector varies in length, while skeleton and camera view representations are fixed.
https://i.imgur.com/QaDksg1.png

This is achieved by the nature of the network design. Specifically, motion encoder uses 1D convolutions with strides, thus output dimensions are proportionally related to the input. On the other hand, the static encoder uses global average pooling in the final layer to produce a fixed-size latent representation:
https://i.imgur.com/Cf7TVKA.png

More detailed design of the encoders and decoder is shown below:
https://i.imgur.com/cpaveFm.png

**Dataset**. Adobe Mixamo is used to obtain sequences of poses of different 3D characters. It allows creating multiple samples where different characters (with different skeleton structure) perform the same motions. These 3D video clips are then projected into 2D by selecting arbitrary view angles and distance to the object. Thus, we can easily create multiple pairs of 2D image sequences of characters (same or different) performing various actions (same or different) from various views.

**Loss functions** used to for training (refer the paper for the detailed formulas):
- *Cross Reconstruction Loss*

 It is a sum of two other losses. The first one is the reconstruction loss where the network tries to reconstruct original input. The second one is cross reconstruction loss where the network tries to reconstruct the sequence where a different character performs the exact same action as the input. It is best shown in the Figure below:
https://i.imgur.com/ewZOAox.png

- *Triplet Loss*
 This loss aims to bring latent spaces of similar motions closer together, while separate apart the ones that are different. It takes two triplets, where each contains two samples that share the same (or very similar) motion and one with different. The same concept is applied to the static latent space.

- Foot velocity loss
 This loss helps to remove the foot skating phenomenon - hands and feet exhibit larger errors that the other keypoints. 
https://i.imgur.com/DclJEde.png
where $V_{global}$ and $V_{joint_n}$ extract the global and local ($n$th joint) velocities from the reconstructed output $\hat{p}_{ij}$, respectively, and map them back to the image units, and $V_{orig_n}$ returns the original global velocity of the $n$th joint from the ground truth, $p_{ij}$

**Normalization**
 - subtract the root position from all joint locations in every frame
 - subtract the mean joint position and divide by the standard deviation (averaged over the entire dataset) 
 - per-frame global velocity is not touched

**Data Augmentation** applied during training:
 - temporal clipping during the batch creation process
 - scaling - same as to use different camera distance to the object
 - flipping symmetrical joints
 - dropping joints to simulate behavior of a real keypoint detector as they often miss some joints
 - adding real video data to the training and use reprojection loss in case no labels are given

**Results and Evaluation** (to be continued) ...

While the summary becomes too long to be a called a summary it is worth mentioning that there are several applications possible with this approach:
 - performance cloning - make any 2D skeleton repeat particular motions
 - motion retrieval - search videos that contain the particular target motion

Lee et al. propose a regularizer to increase the size of linear regions of rectified deep networks around training and test points. Specifically, they assume piece-wise linear networks, in its most simplistic form consisting of linear layers (fully connected layers, convolutional layers) and ReLU activation functions. In these networks, linear regions are determined by activation patterns, i.e., a pattern indicating which neurons have value greater than zero. Then, the goal is to compute, and later to increase, the size $\epsilon$ such that the $L_p$-ball of radius $\epsilon$ around a sample $x$, denoted $B_{\epsilon,p}(x)$ is contained within one linear region (corresponding to one activation pattern). Formally, letting $S(x)$ denote the set of feasible inputs $x$ for a given activation pattern, the task is to determine

$\hat{\epsilon}_{x,p} = \max_{\epsilon \geq 0, B_{\epsilon,p}(x) \subset S(x)} \epsilon$.

For $p = 1, 2, \infty$, the authors show how $\hat{\epsilon}_{x,p}$ can be computed efficiently. For $p = 2$, for example, it results in

$\hat{\epsilon}_{x,p} = \min_{(i,j) \in I} \frac{|z_j^i|}{\|\nabla_x z_j^i\|_2}$.

Here, $z_j^i$ corresponds to the $j$th neuron in the $i$th layer of a multi-layer perceptron with ReLU activations; and $I$ contains all the indices of hidden neurons. This analytical form can then used to add a regularizer to encourage the network to learn larger linear regions:

$\min_\theta \sum_{(x,y) \in D} \left[\mathcal{L}(f_\theta(x), y) - \lambda \min_{(i,j) \in I} \frac{|z_j^i|}{\|\nabla_x z_j^i\|_2}\right]$

where $f_\theta$ is the neural network with paramters $\theta$. In the remainder of the paper, the authors propose a relaxed version of this training procedure that resembles a max-margin formulation and discuss efficient computation of the involved derivatives $\nabla_x z_j^i$ without too many additional forward/backward passes.

https://i.imgur.com/jSc9zbw.jpg
Figure 1: Visualization of locally linear regions for three different models on toy 2D data.

On toy data and datasets such as MNIST and CalTech-256, it is shown that the training procedure is effective in the sense that larger linear regions around training and test points are learned. For example, on a 2D toy dataset, Figure 1 visualizes the linear regions for the optimal regularizer as well as the proposed relaxed version.

Also find this summary at [davidstutz.de](https://davidstutz.de/category/reading/).