Ricci Flow, Geomstats, Lie Groups on Julia & EconoPhysics (and belated Happy Valentines day)

Hey Readers,

Again... thanks for reading my blog. I have been super-busy with lot of work. But,  30,000 views in single night is pretty cool. It is 9 PM in the evening, I just came from my walk in Maryland and I am on a mission to motivate everyone to learn more about data, AI, Math, Topology and so on. I should give more guest lectures when I get time.. but, for now please have a look at wonderful lecture by Geometric Statistics in Machine Learning GeomStats with Nina Miolane - https://www.youtube.com/watch?v=3yNInHuIWlY&t=1251s 

Hence, the next blogpost in the night on Ricci Flow (the topological concept that Grigori Perelman used to prove Poincare Conjecture) and Network Geometry.

1. Network geometry and market instability -https://royalsocietypublishing.org/doi/10.1098/rsos.201734

This paper talks about Time series of log-returns over a 32-year period (1985–2016) with network of stocks with 

1. Ollivier–Ricci (ORE), 
2. Forman–Ricci (FRE), 
3. Menger–Ricci (MRE), 
4. Haantjes–Ricci (HRE) 
5. Algebraic topological aspects, such as the homology groups and Betti numbers

minimum risk Markowitz portfolio of all the stocks, to better understand tipping points, systemic risk and resilience in financial networks, and enable us to develop monitoring tools required for the highly interconnected financial systems and perhaps forecast future financial crises and market slowdowns with use of Python package NetworkX (Yeah.. I used to use a lot of R, SAS, SQL, Cytoscape, Python in 2006 but Julia became my favorite language in 2012).

You can also check out few packages such as 

1. TheanoGeometry ( https://arxiv.org/abs/1712.08364 )- Riemann, Ricci and scalar curvature & geodesics  
2. Geomstats ( https://arxiv.org/abs/2004.04667 )   
3. McTorch  - McTorch, a manifold optimization library for deep learning ( https://arxiv.org/abs/1810.01811 ) 
4. Pymanopt: A Python Toolbox for Manifold Optimization using Automatic Differentiation ( https://arxiv.org/abs/1603.03236 )
5. Topological Entropy for Geodesic Flows under a Ricci Curvature condition: Jacobi field, Ricci curvature, topological entropy, tangent bundle, Negative Ricci curvature, isometry group and Injectivity radius - ( https://www.ams.org/journals/proc/1997-125-06/S0002-9939-97-03780-5/S0002-9939-97-03780-5.pdf )
6. And how can I forget my new favorite Julia for manifold Manifolds.jl: An Extensible Julia Framework for Data Analysis on Manifolds - a fast and easy to use library of Riemannian manifolds and Lie groups - ( https://arxiv.org/pdf/2106.08777.pdf ) 

Ricci Flow, Economics & a brilliant blog by Gita Gopinath

Hey readers,

It's 2:40 AM on Saturday night. Thanks for reading my DataHulk blog. I was glad to see 20,000 views in a single night on my last post - Yang Mills, Vector bundles, Quantum Information Geometry & Fisher-Bures Adversary Graph Convolutional Networks etc. And Yes, I noticed Terence Tao wrote his blog on the same day talking about Sphere Pinching paper (Smart people think alike I guess) - https://terrytao.wordpress.com/2022/02/08/perfectly-packing-a-square-by-squares-of-nearly-harmonic-sidelength/ 

I used to write this blog 12 years back to motivate everyone to learn math, data & AI folks. By the way, I have an exciting blogpost today which might change the way some of my followers look at economics (I know.. I know.. I promised to write about Spin Neural Networks, diffusion and capsule networks in my last blog.. which I will write later). 

1. Ricci Flow & Economics (And why Gita Gopinath's sentence in her blog is genius) - 

Here is an interesting paper followed by 2013-14 Ricci Flow & Poincare Conjecture in topology of Social Networks & Information Geometry article ( https://www.kdnuggets.com/2014/05/poincare-conjecture-perelman-topology-social-networks.html ).... Ricci curvature: An economic indicator for market fragility and systemic risk - https://www.researchgate.net/publication/303600815_Ricci_curvature_An_economic_indicator_for_market_fragility_and_systemic_risk which is an analysis based on geometric feature extraction on network data of daily returns from a set of stocks & market topology comprising the Standard and Poor’s 500 (S&P 500) over a 15-year span to highlight the fact that corresponding changes in Ricci curvature, is negatively correlated to increases in network fragility. 

network, is negatively correlated to increases in network fragility. To illustrate this insight, we examine daily

returns from a set of stocks comprising the Standard and Poor’s 500 (S&P 500) over a 15-year span to highlight

the fact that corresponding changes in Ricci curvature constitute a financial “crash hallmark.”

  

Average Ricci curvature over a 15-year span of the S&P 500 - Choosing a window of T = 22 days, we see that curvature captures several financial crashes and show that, on average, market behavior is fragile.

2. Gita Gopinath - 

Here is an interesting & brilliant blog (April 2021) by Gita Gopinath on divergent recovery -
Managing Divergent Recoveries - https://blogs.imf.org/2021/04/06/managing-divergent-recoveries/ speaking about 

1. Financial risks, 

2. Financial stability risks using macro-prudential tools, 

3. Cross-country gaps, global poverty reduction, 

4. Emerging markets, 

5. Divergent recovery paths, 

6. High degree of uncertainty and developing economies 

7. International liquidity

8. Liquidity protection

9. Debt restructuring 

10. financial stability risks using macro-prudential tools

11. Withdrawn loan payments, firm insolvencies

12. Cross-border profit shifting

which reminded me of network entropy, nodal entropy, geodesics curvature & divergence mentioned in the Sandhu's paper - Ricci curvature & Ricci Flow: An economic indicator for market fragility and systemic risk and my KDnuggets article on which intended to perform Ricci Flow on Social network analysis & information economics. 

Yang Mills, Vector bundles, Quantum Information Geometry & Fisher-Bures Adversary Graph Convolutional Networks etc.

Hello guys,


It's been a long time since my last post & its Tuesday night.. Yes.. I am still DataHulk.. When I get angry I work on Big Data... :) 

- And off course.. Maryna Viazovska (one of the researcher who might win field prize as per my first blogpost based on Graph Mining, NLP, Reinforcement Learning & Profiling along with Bhargav Bhatt etc. - Graph Mining, Network Science + Topology & Predicting Field Prize - https://researchcircle.blogspot.com/2021/12/graph-mining-network-science-topology.html) has published new paper on Hyperbolic Fourier series - https://arxiv.org/abs/2110.00148

....... while Terence Tao has published new paper on Perfectly packing a square by squares of nearly harmonic sidelength - https://arxiv.org/abs/2202.03594 (Another signal that Sphere Packing problem is flavor of the season and Maryna Viazovska might win Field Prize for her effort on Sphere Packing in N = 8 & 24) 

There are 4 papers which I suggest reading everyone who is interested in intersection of Yang Mills, Vector Bundles & Graph neural networks -

1. Fisher-Bures Adversary Graph Convolutional Networks (quantum information geometry & Fisher information of the neural network) -

http://proceedings.mlr.press/v115/sun20a.html

You can enhance this Graph Neural Network further with reconstructing Quantum geometry from Quantum Information from the following paper. 

2. Reconstructing Quantum Geometry from Quantum Information 

https://arxiv.org/pdf/gr-qc/0501075.pdf

2. The Yang-Mills α-flow in vector bundles over four manifolds and its applications (Gang Tian) - 

https://arxiv.org/abs/1303.0628

4. Vector Bundles On Algebraic Varieties (This is the most important paper by Dr. Atiyah & team from 1984 in collaboration with TIFR Mumbai & Oxford University) -

http://www.math.tifr.res.in/~publ/studies/Vector-Bundles-On-Algebraic-Varieties.pdf

One of these days, I will write in detail why Yang mills & vector bundles has potential solution while working on graph neural networks because of Quantum Information Geometry connection. It's going to be one hell of an orchestration. 

For now, look at the beauty of Sir Atiyah's 1984 paper in collaboration with TIFR. (BTW, Update 8 days after this blogpost was published.. on 17th Feb Sir Atiyah's Student George Lusztig won Wolf Prize for representation theory & Quantum Groups) 

My next blog is going to be about Spin Networks.    

Federated learning & Security (Story of a CEO & 2 researchers)

Hello Followers & Data Friends,

Again, thanks to someone who wrote this in my book review and helped me to restart this blog again.. I am glad someone still remembers my old blog and so many got motivated to enter AI & Data Science after reading it. My humble thank you - 

It's Sunday night (11:30 PM) and amazing times that we live in where AI and Data is the new oil. And it's also the time to have a deep look at Sundar's last month's interview where he spoke about Federated Learning (starting at 27:50). (Yeah, it was another busy weekend with lot of work but I still listen to Sundar's speeches on my walk) - https://www.youtube.com/watch?v=EuF8nv53JeI&t=1997s

Federated Learning & Edge Computing is an evolving topic most popular since Google announced its cookie-less marketing world last year - https://www.gartner.com/en/marketing/insights/articles/three-steps-for-marketers-to-prepare-for-a-cookieless-world

Sundar also spoke about Privacy, AI and a lot of futuristic things.

The books/papers that came to my mind immediately after listening to the interview are -

1. Milind Tambe's book on Security & Game Theory (2011) - https://teamcore.seas.harvard.edu/publications/security-and-game-theory-algorithms-deployed-systems-lessons-learned

This is one game theory book I liked a lot in 2011 and wrote about it in my blog that time. It was fresh with good perspective security & game theory while it was written in a simple language. Also, amazing amazing chapter 12 on Stackelberg versus Nash in Security Games. Every game theory & security student should read this chapter and if possible attend Milind Tambe's lectures. His lectures have amazing clarity & awesome vision. (Yes. I don't promote my own books 😜)

2. Advances and Open Problems in Federated Learning (2019) - https://arxiv.org/abs/1912.04977 by Ramesh Raskar & the team. A wonderful wonderful paper on Federated Learning. 

The chapter I Like in this paper/booklet is - Adapting ML Workflows for Federated Learning, Hyperparameter Tuning and Neural Architecture Search (NAS) which is inspired by Chaoyang He, Murali Annavaram, and Salman Avestimehr' s paper on FEDNAS - Federated deep learning via neural architecture search. 

I will try to cover these things in my any upcoming guest lectures very soon. I have been extremely busy. For now, it's time to listen to Milind Tambe's all lectures from the past including AI for Social Good lecture including his JPMorgan lecture... 

More coming soon on this topic...

4 interesting papers on Ricci Flow & Deep Learning

Hey Folks,

Greetings. Its Saturday night and a new blog post from DataHulk.. Funny Funny name.. I came up with this name 12 years back .. when I get angry I work on data...! 

It's been some time since I spoke about Neural Networks, GANs & Ricci Flow (Ricci Flow was an important geometric tool in solving Poincare Conjecture....). Please feel free to check out latest paper on Shannon entropy power on Riemannian manifolds and Ricci flow - https://arxiv.org/pdf/2001.00410.pdf (Lot of hints to Information Geometry & Topology..) 

Here are 3 interesting papers which came out since my talks & articles (apart from Melanie Weber's papers at Princeton on Big Data & Ricci Flow.

1. Zhejiang University - THOUGHTS ON THE CONSISTENCY BETWEEN RICCI FLOW AND NEURAL NETWORK BEHAVIOR - https://arxiv.org/pdf/2111.08410.pdf

2. Oxford University - Over-squashing, Bottlenecks, and Graph Ricci curvature - https://towardsdatascience.com/over-squashing-bottlenecks-and-graph-ricci-curvature-c238b7169e16
3. University of Cambridge - RicciNets: Curvature-guided Pruning of High-performance Neural Networks Using Ricci Flow - https://arxiv.org/abs/2007.04216 

 

The computational graph is pruned based on a node mass probability function defined by local graph measures and weighted by hyperparameters produced by a reinforcement learning-based controller neural network. 

3 unique papers with 3 different ideas of using Ricci flow in deep learning. 

The 4th one which is very very unique is CONVERGENCE OF RICCI FLOW SOLUTIONS TO TAUB-NUT by Francesco de Giovanni - https://arxiv.org/pdf/2008.03969.pdf  

It does not show the applications through deep learning yet but as amazing paper which might have some future applications in deep learning.

Interesting case of Schrodinger equation & deep learning (and possible application in Commutative Hodge Conjecture)

If my readers remember my old blog (written under the pseudoname of DataHulk 😏), I used to talk about Physics Informed Neural Networks, Physics formed deep learning & Data, Machine learning & Deep learning to solve math conjectures 12 years back... much before The Ramanujan Machine (AI to Solve Math Conjectures) was born. 

And Yes, I saw review on my 2013 book - Healthcare Social Media Management and Analytics, so I know some of my readers have missed me 🙌🙌🙌





1. Here is a curious case which was written few months back - Data-driven vector soliton solutions of coupled nonlinear Schrödinger equation using a deep learning algorithm - https://www.researchgate.net/publication/355789067_Data-driven_vector_soliton_solutions_of_coupled_nonlinear_Schrodinger_equation_using_a_deep_learning_algorithm In this paper, there is pre-fixed multi-stage training algorithm by combining the error measurement & multi-stage training. This algorithm is much better suited for different dynamical behaviors of solitons with faster convergence rate.


2. Recently, G. Tabuada from MIT proposed a series of noncommutative counterparts all conjectures including Grothendieck standard conjecture, Voevodsky nilpotence conjecture, Tate conjecture, Weil conjecture etc. XUN LIN has also proposed NON-COMMUTATIVE HODGE CONJECTURE.


Similar physics informed neural nets or GANs can be used for commutative Hodge Conjecture. More to come soon on this topic.


On a different topic, Please do listen to András Juhász & Marc Lackenby (similar to University of Sydney mathematician Geordie Williamson's work with DeepMind on representation theory) - https://www.youtube.com/watch?v=hIUiPi-jAjM

Swarm Intelligence - Another curious case of Duck Swarm algorithm & New Google algorithm on Pathway Analysis

Hey guys, Thanks for reading my restarted blog. A lot of views and reads (that also on 31st night.. I must say I have very cool & eager readers.. I hope everyone had a great new year's night..  

Math is everywhere.. A lot of my readers might remember my old orchestration on Ant Colony Optimization, Swarm Intelligence & Quantum-behaved Particle Swarm Optimization algorithm (And a fun hint on How Ant Colony Optimization is used in CyberSecurity with reference to first Avengers movie dialogue - Nick Fury to Loci when he is in the prison- "the touch of the ant and the click" ...

And And And... funnier hint that Marvel might release a movie on Ant-Man etc..

A lot of advances have happened in Swarm Intelligence since then.... 

1. Quantum Particle swarm optimization (PSO), 

2. Firefly algorithm (FA), 

3. Chicken swarm optimization (CSO), 

4. Grey wolf optimizer (GWO), 

5. Sine cosine algorithm (SCA), 

6. Marine-predators algorithm (MPA)

7. Archimedes optimization algorithm (AOA)

8. And Surprise Surprise - Duck Swarm Algorithm 

Please check out 2021 paper - Duck swarm algorithm: a novel swarm intelligence algorithm by Zhang , Wen, Yang - South China University of Technology, Guangzhou 

A couple of simple images from the paper - 

And the Pathway Analysis based on duck swarm - 

By the way, A very very interesting thing happened a few months back. Google came up with their own AI tool on pathway analysis - https://blog.google/technology/ai/introducing-pathways-next-generation-ai-architecture/  

The question is would Google use more & more swarm intelligence in their pathway analysis?