General Topology Pdf
Mirthe Luria
The first big challenge I faced is when approaching William Boothby's An Introduction to Differentiable Manifolds and Riemannian Geometry. I soon realized that I needed to learn some algebraic topology and differential topology, which I did much later. Nevertheless, everyday topology for me is still mostly general topology. I could say every bits and pieces of Munkres's Part I has its use in analysis, but hell, its really a lot to memorize.
I read the book through, or some chapters again and again. But somehow I still cannot memorize everything. So as a result, I had to come back to Munkres from time to time, the only difference being now I know what I am looking for. But I definitely cannot say I learn topology very well. This has puzzled me for a long time, because usually after I read a book three times, I can have a good feeling of at least the big picture. But with Munkres, its just less organized in my mind, not the big blocks (connected/ compactness/ countability/ separation/ compactification/ metrization/ completeness/ Baire space), but those small yet useful lemma/theorems/corollaries.
A very good book for point set topology which emphasizes the connections with analysis and which is cheap is Albert Wilansky's ironically but appropriately titled Topology For Analysis.The book is somewhat more advanced then Munkres, it assumes the student has a good working understanding of the basic topology of Euclidean and metric spaces from undergraduate analysis. Wilansky's book begins with convergence,reviewing basic sequences and proceeding to develop general convergence via nets and filters. This sets the stage for developing all the topological machinery needed for functional analysis and operator theory, which I think is what you want. It discusses semimetrics and norms, separation axioms, compactification, function spaces and uniform spaces, as well as a number of topics that usually reserved for functional analysis courses, such as the weak topology,topological groups and the Gleason map. I think you'll find this book quite helpful for getting the topological structure of analysis mastered-and best of all, it's cheap.
If you're interested by algebraic topology, the book of Fulton or the book of Bott and Tu are using differential forms and motivate some results by analytic approach. In the same spirit you can take a look to the fantastic book From Calculus to Cohomology.
- Springer Shop
- Amazon.com
- Barnes&Noble.com
- Books-A-Million
- IndieBound
- Find in a library
- All sellers
As a technique to investigate link-level loss rates of a computer network with low operational cost, loss tomography has received considerable attentions in recent years. A number of parameter estimation methods have been proposed for loss tomography of networks with a tree structure as well as a general topological structure. However, these methods suffer from either high computational cost or insufficient use of information in the data. In this paper, we provide both theoretical results and practical algorithms for parameter estimation in loss tomography. By introducing a group of novel statistics and alternative parameter systems, we find that the likelihood function of the observed data from loss tomography keeps exactly the same mathematical formulation for tree and general topologies, revealing that networks with different topologies share the same mathematical nature for loss tomography. More importantly, we discover that a reparametrization of the likelihood function belongs to the standard exponential family, which is convex and has a unique mode under regularity conditions. Based on these theoretical results, novel algorithms to find the MLE are developed. Compared to existing methods in the literature, the proposed methods enjoy great computational advantages.
Usually neural networks consist from layers, but is there research effort that tries to investigate more general topologies for connections among neurals, e.g. arbitrary directed acyclic graphs (DAGs).
But I have no idea, which answer is the correct one. Reading the answer on -neural-network-with-arbitrary-topology I start to think that answer 1 is the correct one, but there is no reference provided.
If answer 3 is correct, then big revolution can be expected. E.g. layered topologies in many cases reduces learning to the matrix exponentiation and good tools for this are created - TensorFlow software and dedicated processors. But there seems to be no software or tools for general topologies is they have some sense indeed.
Artificial network topologies are generally cyclic, not acyclic in terms of their causality or their signal pathways, depending on how you depict them theoretically. These are three basic examples from among dozens in the literature and in the open source repositories.
Back-propagation represents the introduction of a deliberate cycle in signal paths in a basic multilayer perceptron, making that topology a sequence of layers represented by vertices, connected sequentially by a set of directed edges representing forward propagation, and a set of directed edges in the reverse direction to distribute the corrective error determined at the network output according to the principle of gradient descent. For efficiency, the corrective signal is distributed recursively backward through the layers to the $N - 1$ matrices of parameters attenuating the signals between $N$ layers. Back propagation requires the formation of these $N - 1$ cycles for convergence to occur.
In attention based networks being touted as theoretically advantageous over LSMT (which has been dominating over CNNs) has a much more complex topology and more cycles above those in supervisory layer than those in GANs.
Turing was aware that his punched tape model was not the best general purpose, high speed computing architecture. He was not intending to prove anything about computing speed but rather what could be computed. His Turing machine had a trivial topology deliberately. He wanted to illustrate his completeness theorem to others and resurrect the forward movement of rationalism after Gödel disturbed it with his two incompleteness theorems.
Similarly, John von Neumann proposed his computing architecture, with a central processing unit (CPU) and unified data and instruction bus, to reduce the number of relays or vacuum tubes, not to maximize parallel algorithm execution. That topology as a directed graph has the instruction controller and the arithmetic unit in the center and everything else branching out from the data and address bus leading from them.
That a topology can accomplish a task is no longer a justification for persisting in the use of that topology, which is why Intel acquired Nirvana, which deviates from traditional von Neumann architecture, DSP architecture, and the current CUDA core architecture that NVidia GPUs use and offer for artificial network realization through C libraries that can be called via integrated Java and Python adapters.
General topologies exist, the most economically viable of which is the CUDA cores begun by NVidia, which can be configured for MLPs, CNNs, RNNs, and general 2D and 3D video processing. They can be configured with or without cycles depending on the characteristics of the parallelism desired.
FORTRAN began to dominate over LISP during the time when general purpose programming began to emerge in many corporations. That is not surprising because humans communicate in orthogonal ways. It is cultural. When a child scribbles, teachers are indoctrinated to say nice things but respond by drawing a shape. If the child draws a square, the teacher smiles. The child is given blocks. The books are rectangular. Text is justified into rectangles.
General topologies exist and are promising and researchers are ready to work with them. It is enthusiasts that can have a dismissive attitude. They don't yet understand the demos they've downloaded and painstakenly tweaked to run on their computer, they're about to launch their AI carrier amidst the growing demand from all the media hype, and now someone is introducing something interesting and not yet implemented in code. The motivational direction is generally to either dismiss or resist the creative proposals.
In this case, Google, CalTech, IBM, MIT, U Toronto, Intel, Tesla, Japan, and a thousand other governments, institutions, corporations, and open source contributors will solve that problem, provided people keep talking about topology and the restrictions inherent in purely Cartesian thinking.
There has been some confusion in terms. The SO reference in the question is an example of thinking that changing an array dimension is changing the topology. If such were so, then there would be no change one could make to the geometry of an AI system that would not be topological. Topology can only have meaning if there are features that are not topological. When one draws a layer, they don't need to increase the height of the rectangle representing it if the number of activitations, the width of the layer, is changed from 100 to 120.
I've also seen academic papers that called the texture or roughness of an error surface its topology. That completely undermines the concept of topology. They meant to use the term topography. Unfortunately neither the publisher nor the editor noticed the error.
aa06259810