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We want to find two matrices $W$ and $H$ such that $V = WH$. Often a goal is to determine underlying patterns in the relationships between the concepts represented by each row and column. $W$ is some $m$ by $n$ matrix and we want the inner dimension of the factorization to be $r$. So $$\underbrace{V}_{m \times n} = \underbrace{W}_{m \times r} \underbrace{H}_{r \times n}$$ Let's consider an example matrix where of three customers (as rows) are associated with three movies (the columns) by a rating value. $$ V = \left[\begin{array}{c c c} 5 & 4 & 1 \\\\ 4 & 5 & 1 \\\\ 2 & 1 & 5 \end{array}\right] $$ We can decompose this into two matrices with $r = 1$. First lets do this without any non-negative constraint using an SVD reshaping matrices based on removing eigenvalues: $$ W = \left[\begin{array}{c c c} -0.656 \\\ -0.652 \\\ -0.379 \end{array}\right], H = \left[\begin{array}{c c c} -6.48 & -6.26 & -3.20\\\\ \end{array}\right] $$ We can also decompose this into two matrices with $r = 1$ subject to the constraint that $w_{ij} \ge 0$ and $h_{ij} \ge 0$. (Note: this is only possible when $v_{ij} \ge 0$): $$ W = \left[\begin{array}{c c c} 0.388 \\\\ 0.386 \\\\ 0.224 \end{array}\right], H = \left[\begin{array}{c c c} 11.22 & 10.57 & 5.41 \\\\ \end{array}\right] $$ Both of these $r=1$ factorizations reconstruct matrix $V$ with the same error. $$ V \approx WH = \left[\begin{array}{c c c} 4.36 & 4.11 & 2.10 \\\ 4.33 & 4.08 & 2.09 \\\ 2.52 & 2.37 & 1.21 \\\ \end{array}\right] $$ If they both yield the same reconstruction error then why is a non-negativity constraint useful? We can see above that it is easy to observe patterns in both factorizations such as similar customers and similar movies. `TODO: motivate why NMF is better` #### Paper Contribution This paper discusses two approaches for iteratively creating a non-negative $W$ and $H$ based on random initial matrices. The paper discusses a multiplicative update rule where the elements of $W$ and $H$ are iteratively transformed by scaling each value such that error is not increased. The multiplicative approach is discussed in contrast to an additive gradient decent based approach where small corrections are iteratively applied. The multiplicative approach can be reduced to this by setting the learning rate ($\eta$) to a ratio that represents the magnitude of the element in $H$ to the scaling factor of $W$ on $H$. ### Still a draft |
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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) |
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Problem ---------- Given an unconstrained image, estimate: 1. 3d pose of human skeleton 2. 3d body mesh Contributions ----------- 1. full body mesh extraction from image 2. improvement of state of the art Datasets ------------- 1. Leeds Sports 2. HumanEva 3. Human3.6M Approach ---------------- Consider the problem both bottom-up and top-down. 1. Bottom-up: DeepCut cnn model to fit joints 2d positions onto the image. 2. top-down: A skinned multi-person linear model (SMPL) is fitted and projected onto 2d joint positions and image. |
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Deeper networks should never have a higher **training** error than smaller ones. In the worst case, the layers should "simply" learn identities. It seems as this is not so easy with conventional networks, as they get much worse with more layers. So the idea is to add identity functions which skip some layers. The network only has to learn the **residuals**. Advantages: * Learning the identity becomes learning 0 which is simpler * Loss in information flow in the forward pass is not a problem anymore * No vanishing / exploding gradient * Identities don't have parameters to be learned ## Evaluation The learning rate starts at 0.1 and is divided by 10 when the error plateaus. Weight decay of 0.0001 ($10^{-4}$), momentum of 0.9. They use mini-batches of size 128. * ImageNet ILSVRC 2015: 3.57% (ensemble) * CIFAR-10: 6.43% * MS COCO: 59.0% mAp@0.5 (ensemble) * PASCAL VOC 2007: 85.6% mAp@0.5 * PASCAL VOC 2012: 83.8% mAp@0.5 ## See also * [DenseNets](http://www.shortscience.org/paper?bibtexKey=journals/corr/1608.06993) |
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An attention mechanism and a separate encoder/decoder are two properties of almost every single neural translation model. The question asked in this paper is- how far can we go without attention and without a separate encoder and decoder? And the answer is- pretty far! The model presented preforms just as well as the attention model of Bahdanau on the four language directions that are studied in the paper. The translation model presented in the paper is basically a simple recurrent language model. A recurrent language model receives at every timestep the current input word and has to predict the next word in the dataset. To translate with such a model, simply give it the current word from the source sentence and have it try to predict the next word from the target sentence. Obviously, in many cases such a simple model wouldn't work. For example, if your sentence was "The white dog" and you wanted to translate to Spanish ("El perro blanco"), at the 2nd timestep, the input would be "white" and the expected output would be "perro" (dog). But how could the model predict "perro" when it hasn't seen "dog" yet? To solve this issue, we preprocess the data before training and insert "empty" padding tokens into the target sentence. When the model outputs such a token, it means that the model would like to read more of the input sentence before emitting the next output word. So in the example from above, we would change the target sentence to "El PAD perro blanco". Now, at timestep 2 the model emits the PAD symbol. At timestep 3, when the input is "dog", the model can emit the token "perro". These padding symbols are deleted in post-processing, before the output is returned to the user. You can see a visualization of the decoding process below: https://i.imgur.com/znI6xoN.png To enable us to use beam search, our model actually receives the previous outputted target token in addition to receiving the current source token at every timestep. PyTorch code for the model is available at https://github.com/ofirpress/YouMayNotNeedAttention |