Computer Vision or CV can be defined as a field of study that aims to develop techniques to enable computers to "see" or develop "vision" and also understand the content of digital images such as photographs and videos. Images and text are all around us these days, and they encircle human society. Smartphones these days have cameras that can capture high-resolution images in just a touch. Sharing photos and videos have never been easier, thanks to social media platforms like Instagram and Facebook. Even with messaging apps like Whatsapp and Telegram, connectivity today has become much easier, and hence it also seems to be getting even simplified day by day.

The implied semantics of direct manipulation is that when a user drags an UI element (in this case, an axis handle), they are signaling to the system that they wished that the corresponding data point had been projected to the location where the UI element was dropped, rather than where it was dragged from. In our case the overall projection is a rotation (originally determined by the Grand Tour), and an arbitrary user manipulation might not necessarily generate a new projection that is also a rotation. Our goal, then, is to find a new rotation which satisfies the user request and is close to the previous state of the Grand Tour projection, so that the resulting state satisfies the user request. In a nutshell, when user drags the ithi {th}ith axis handle by (dx,dy)(dx, dy)(dx,dy), we add them to the first two entries of the ithi {th}ith row of the Grand Tour matrix, and then perform Gram-Schmidt orthonormalization on the rows of the new matrix. Rows have to be reordered such that the ithi {th}ith row is considered first in the Gram-Schmidt procedure.

Collaborative Innovation Center of Quantum Matter, Beijing, 100084, China (Dated: March 12, 2018) Motivated by the question that whether the empirical fitting of data by neural networks can yield the same structure of physical laws, we apply neural networks to a quantum mechanical two-body scattering problem with short-range potentials--a problem by itself plays an important role in many branches of physics. After training, the neural network can accurately predict s - wave scattering length, which governs the low-energy scattering physics. By visualizing the neural network, we show that it develops perturbation theory order by order when the potential depth increases, without solving the Schr odinger equation or obtaining the wavefunction explicitly. The result provides an important benchmark to the machine-assisted physics research or even automated machine learning physics laws. Human physicists have made great achievements in discovering laws of physics during the last several centuries.

Artificial neural networks became very popular in recent years, mostly because of their success in tasks of image and speech recognition. While research in this area started more than 60 years ago and many different network architectures were developed during first decades of research, the only architecture that became popular in applications is MLP (multilayer perceptron) -- parametrized multilayer functions trained (optimized) with variations of gradient descent. Later based on MLP approach of training application-specific architectures emerged (such as convolutional and recurrent networks). Probably it is a good idea to understand the behavior of a neural network by visualizing it. While dependencies modelled in machine learning, in particular by neural networks, are multidimensional, we are limited in our visualization abilities to three dimensions.

So the question I have is: what does the frontier of the space of optimal networks look like, what are the inherent limits of depth vs expressivity of these models, and are there dimensional scaling laws that can describe all this in an information theoretic way? This recent paper gives a great treatment on the expressivity of convolution networks by using a deep layered architecture that generalizes convolutional neural networks called sim-nets. As a simple first step I wanted to see what could be done to visualize the operations a deep neural net performs. So I constructed a standard network that takes vector inputs of size 2 and produces vector outputs of size 3 which we can think of as a mapping of the cartesian plane into RGB color space. Taking many copies of this net and randomly initializing them, (with normally distributed weights and biases) we can plot them in a grid and see the networks' outputs as a set of images.