This paper introduces a bunch of tricks which make sense for visual tracking. These tricks are as followed:
1. At test time, a crop with center at the previous frame's bounding box's center with size larger than the bounding box is given along with the search area in the current frame.
2. Training offline on a large set of videos (where object bounding boxes are given for a subset of frames) and images with object bounding boxes.
3. Network takes two images: i) a crop of the image/frame around the bounding box and  ii) the image centered at the center of the bounding box. Given the later, network regresses the bounding box in i).
4. Above crops are sampled such that the ground truth bounding box center in i) is not very far from the center in ii), hence network prefers smooth motion.

My take: This is very nice way to use still images to train image correlation task and hence can be used for tracking. Speed on gpu is very impressive but still not comparable on CPUs.

Xu and Mannor provide a theoretical paper on robustness and generalization where their notion of robustness is based on the idea that the difference in loss should be small for samples that are close. This implies that, e.g., for a test sample close to a training sample, the loss on both samples should be similar. The authors formalize this notion as follows:

Definition: Let $A$ be a learning algorithm and $S \subset Z$ be a training set such that $A(S)$ denotes the model learned on $S$ by $A$; the algorithm $A$ is $(K, \epsilon(S))$-robust if $Z$ can be partitioned into $K$ disjoint sets, denoted $C_i$ such that $\forall s \in S$ it holds:

$s,z \in C_i \rightarrow |l(A(S), s) – l(A(S), z)| \leq \epsilon(S)$.

In words, this means that we can partition the space $Z$ (which is $X \times Y$ for a supervised problem) into a finite set of subsets and whenever a sample falls into the same partition as a training sample, the learned model should have nearly the same loss on both samples. Note that this notion does not entirely match the notion of adversarial robustness as commonly referred to nowadays. The main difference is that the partition can be chosen, while for adversarial robustness, the “partition” (usually in form of epsilon-balls around training and testing samples) is fixed.

Based on the above notion of robustness, the authors provide PAC bounds for robust algorithms, i.e. generalization performance of $A$ is linked to its generalization. Furthermore, in several examples, common machine learning techniques such as SVMs and neural networks are shown to be robust under specific conditions. For neural networks, for example, an upper bound on the $L_1$ norm of weights and the requirement of Lipschitz continuity is enough. This actually related to work on adversarial robustness, where Lipschitz continuity and weight regularization is also studied.

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

This summary builds extensively on my prior summary of SIRENs, so if you haven't read that summary or the underlying paper yet, I'd recommend doing that first! 

At a high level, the idea of SIRENs is to use a neural network to learn a compressed, continuous representation of an image, where the neural network encodes a mapping from (x, y) to the pixel value at that location, and the image can be reconstructed (or, potentially, expanded in size) by sampling from that function across the full range of the image. To do this effectively, they use sinusoidal activation functions, which let them match not just the output of the neural network f(x, y) to the true image, but also the first and second derivatives of the neural network to the first and second derivatives of the true image, which provides a more robust training signal. 

NERFs builds on this idea, but instead of trying to learn a continuous representation of an image (mapping from 2D position to 3D RGB), they try to learn a continuous representation of a scene, mapping from position (specified with with three coordinates) and viewing direction (specified with two angles) to the RGB color at a given point in a 3D grid (or "voxel", analogous to "pixel"), as well as the *density* or opacity of that point. 

Why is this interesting? Because if you have a NERF that has learned a good underlying function of a particular 3D scene, you can theoretically take samples of that scene from arbitrary angles, even angles not seen during training. It essentially functions as a usable 3D model of a scene, but one that, because it's stored in the weights of a neural network, and specified in a continuous function, is far smaller than actually storing all the values of all the voxels in a 3D scene (the authors give an example of 5MB vs 15GB for a NERF vs a full 3D model). To get some intuition for this, consider that if you wanted to store the curve represented by a particular third-degree polynomial function between 0 and 10,000 it would be much more space-efficient to simply store the 3 coefficients of that polynomial, and be able to sample from it at your desired granularity at will, rather than storing many empirically sampled points from along the curve. 

https://i.imgur.com/0c33YqV.png

How is a NERF model learned?

- The (x, y, z) position of each point is encoded as a combination of sine-wave, Fourier-style curves of increasingly higher frequency. This is similar to the positional encoding used by transformers. In practical turns, this means a location in space will be represented as a vector calculated as [some point on a low-frequency curve, some point on a slightly higher frequency curve..., some point on the highest-frequency curve]. This doesn't contain any more *information* than the (x, y, z) representation, but it does empirically seem to help training when you separate the frequencies like this
- You take a dataset of images for which viewing direction is known, and simulate sending a ray through the scene in that direction, hitting some line (or possibly tube?) of voxels on the way. You calculate the perceived color at that point, which is an integral of the color information and density/opacity returned by your model, for each point. Intuitively, if you have a high opacity weight early on, that part of the object blocks any voxels further in the ray, whereas if the opacity weight is lower, more of the voxels behind will contribute to the overall effective color perceived. You then compare these predicted perceived colors to the actual colors captured by the 2D image, and train on the prediction error.
- (One note on sampling: the paper proposes a hierarchical sampling scheme to help with sampling efficiently along the ray, first taking a course sample, and then adding additional samples in regions of high predicted density)
- At the end of training, you have a network that hopefully captures the information from *that particular scene*. A notable downside of this approach is that it's quite slow for any use cases that require training on many scenes, since each individual scene network takes about 1-2 days of GPU time to train

In the two years since it’s been introduced, the Transformer architecture, which uses multi-headed self-attention in lieu of recurrent models to consolidate information about input and output sequences, has more or less taken the world of language processing and understanding by storm. It has become the default choice for language for problems like translation and questioning answering, and was the foundation of OpenAI’s massive language-model-trained GPT. In this context, I really appreciate this paper’s work to try to build our collective intuitions about the structure, specifically by trying to understand how the multiple heads that make up the aforementioned multi-head attention divide up importance and specialize function. 

As a quick orientation, attention works by projecting each value in the sequence into query, key, and value vectors. Then, each element in the sequence creates its next-layer value by calculating a function of query and key (typically dot product) and putting that in a softmax against the query results with every other element. This weighting distribution is then used as the weights of a weighted average, combining the values together. By default this is a single operation, with a single set of projection matrices, but in the Transformer approach, they use multi-headed attention, which simply means that they learn independent parameters for multiple independent attention “filters”, each of which can then notionally specialize to pull in and prioritize a certain kind of information. 
https://i.imgur.com/yuC91Ja.png

The high level theses of this paper are: 
- Among attention heads, there’s a core of the most crucial and important ones, and then a long tail of heads that can be pruned (or, have their weight in the concatenation of all attention heads brought to nearly zero) and have a small effect on performance 
- It’s possible and useful to divide up the heads according to the kinds of other tokens that it is most consistently pulling information from. The authors identify three: positional, syntactic, and rare.  Positional heads consistently (>90% of the time) put their maximum attention weight on the same position relative to the query word. Syntactic heads are those that recurringly in the same grammatical relation to the query, the subject to its verb, or the adjective to its noun, for example. Rare words is not a frequently used head pattern, but it is a very valuable head within the first layer, and will consistently put its highest weight on the lowest-frequency word in a given sentence. 

An interesting side note here is that the authors tried at multiple stages in pruning to retrain a network using only the connections between unpruned heads, and restarting from scratch. However, in a effect reminiscent of the Lottery Ticket Thesis, retraining from scratch cannot get quite the same performance. 

This paper came on my radar after winning Best Paper recently at ICLR, and all in all I found it a clever way of engineering a somewhat complicated inductive bias into a differentiable structure. The empirical results weren’t compelling enough to suggest that this structural shift made a regime-change difference in performing, but it does seem to have some consistently stronger ability to do syntactic evaluation across large gaps in sentences. 

The core premise of this paper is that, while language is to some extent sequence-like, it is in a more fundamental sense tree-like: a recursive structure of modified words, phrases, and clauses, aggregating up to a fully complete sentence. In practical terms, this cashes out to parse trees, labels akin to the sentence diagrams that you or I perhaps did once upon a time in grade school. 

https://i.imgur.com/GAJP7ji.png

Given this, if you want to effectively model language, it might be useful to have a neural network structure explicitly designed to track where you are in the tree. To do this, the authors of this paper use a clever activation function scheme based on the intuition that you can think of jumping between levels of the tree as adding information to the stack of local context, and then removing that information from the stack when you’ve reached the end of some local phrase. In the framework of a LSTM, which has explicit gating mechanisms for both “forgetting” (removing information from cell memory) and input (adding information to the representation within cell memory) this can be understood as forcing a certain structure of input and forgetting, where you have to sequentially “close out” or add nodes as you move up or down the tree. 


To represent this mathematically, the authors use a new activation function they developed, termed cumulative max or cumax. In the same way that the softmax is a differentiable (i.e. “soft”) version of an argmax, the cumulative max is a softened version of a vector that has zeros up to some switch point k, and ones thereafter. If you had such a vector as your forget mask, then “closing out” a layer in your tree would be equivalent to shifting the index where you switch from 0 to 1 up by one, so that a layer that previously had a “remember” value of 1.0 now is removing its content from the stack.  However, since we need to differentiate, this notional 0/1 vector is instead represented as a cumulative sum of a softmax, which can be thought of as the continuous-valued probability that you’ve reached that switch-point by any given point in the vector. Outside of the abstractions of what we’re imagining this cumax function to represent, in a practical sense, it does strictly enforce that you monotonically remember or input more as you move along the vector. This has the practical fact that the network will be biased towards remembering information at one end of the representation vector for longer, meaning it could be a useful inductive bias around storing information that has a more long-term usefulness to it. One advantage that this system has over a previous system that, for example, had each layer of the LSTM operate on a different forgetting-decay timescale, is that this is a soft approximation, so, up to the number of neurons in the representation, the model can dynamically approximate whatever number of tree nodes it likes, rather than being explicitly correspondent with the number of layers. 

Beyond being a mathematically clever idea, the question of whether it improves performance is a little mixed. It does consistently worse at tasks that require keeping track of short term dependency information, but seems to do better at more long-term tasks, although not in a perfectly consistent or overly dramatic way. My overall read is that this is a neat idea, and I’m interested to see if it gets built on, as well as interested to see later papers that do some introspective work to validate whether the model is actually using this inductive bias in the tree-like way that we’re hoping and imagining it will.