This document provides an overview of hierarchical representation with hyperbolic geometry. It introduces hyperbolic space as an alternative to Euclidean space for embedding symbolic and hierarchical data. Key points covered include: (1) the limitations of Euclidean embedding for graph structures, (2) definitions of hyperbolic space and the Poincare disk model, (3) optimization techniques for gradient descent in hyperbolic space including calculating gradients and using retractions, and (4) simple toy experiments demonstrating optimization in hyperbolic space.
One of the main reasons for the popularity of Dijkstra's Algorithm is that it is one of the most important and useful algorithms available for generating (exact) optimal solutions to a large class of shortest path problems. The point being that this class of problems is extremely important theoretically, practically, as well as educationally.
The following presentation is an introduction to the Algebraic Methods โ part one for level 4 Mathematics. This resources is a part of the 2009/2010 Engineering (foundation degree, BEng and HN) courses from University of Wales Newport (course codes H101, H691, H620, HH37 and 001H). This resource is a part of the core modules for the full time 1st year undergraduate programme.
The BEng & Foundation Degrees and HNC/D in Engineering are designed to meet the needs of employers by placing the emphasis on the theoretical, practical and vocational aspects of engineering within the workplace and beyond. Engineering is becoming more high profile, and therefore more in demand as a skill set, in todayโs high-tech world. This course has been designed to provide you with knowledge, skills and practical experience encountered in everyday engineering environments.
with today's advanced technology like photoshop, paint etc. we need to understand some basic concepts like how they are cropping the image , tilt the image etc.
In our presentation you will find basic introduction of 2D transformation.
Shortest path search for real road networks and dynamic costs with pgRoutingantonpa
ย
This presentation will show the inside and current state of pgRouting
development. It will explain the shortest path search in real road
networks and how the data structure is important for getting better
routing results. We will show how you can improve the quality of the search with dynamic costs and make the result look closer to the reality. We will demonstrate the way of using pgRouting together with other Open Source tools. Also you will learn about difficulties and limitations of implementing routing functionality in GIS applications, the difference between algorithms and their performance.
pgRouting is an extension of PostgreSQL and PostGIS. A predecessor of
pgRouting - pgDijkstra, written by Sylvain Pasche from Camptocamp, was
extended by Orkney (Japan) and renamed to pgRouting, which now is a part of the PostLBS project.
pgRouting can perform:
* shortest path search (with 3 different algorithms)
* Traveling Salesperson Problem solution (TSP)
* driving distance geometry calculation
This Presentation Is Specially Made For Those Engineering Students Who are In Gujarat Technological University. This Presentation Clears Your All Doubts About Basics Fundamentals of Numerical Integration. Also You Will Learn Different Types Of Error Formula To Solve the Numerical Integration Sum.
Here we have included details about relaxation method and some examples .
Contribution - Parinda Rajapakha, Hashan Wanniarachchi, Sameera Horawalawithana, Thilina Gamalath, Samudra Herath and Pavithri Fernando.
This presentation covers scalar quantity, vector quantity, addition of vectors & multiplication of vector. I hope this PPT will be helpful for Instructors as well as students.
Concepts and Applications of the Fundamental Theorem of Line Integrals.pdfJacobBraginsky
ย
A three-part examination of the Fundamental Theorem of Line Integrals. Learn how to use this theorem in multivariable calculus. Simplify the process of solving line integrals using the FTLI.
One of the main reasons for the popularity of Dijkstra's Algorithm is that it is one of the most important and useful algorithms available for generating (exact) optimal solutions to a large class of shortest path problems. The point being that this class of problems is extremely important theoretically, practically, as well as educationally.
The following presentation is an introduction to the Algebraic Methods โ part one for level 4 Mathematics. This resources is a part of the 2009/2010 Engineering (foundation degree, BEng and HN) courses from University of Wales Newport (course codes H101, H691, H620, HH37 and 001H). This resource is a part of the core modules for the full time 1st year undergraduate programme.
The BEng & Foundation Degrees and HNC/D in Engineering are designed to meet the needs of employers by placing the emphasis on the theoretical, practical and vocational aspects of engineering within the workplace and beyond. Engineering is becoming more high profile, and therefore more in demand as a skill set, in todayโs high-tech world. This course has been designed to provide you with knowledge, skills and practical experience encountered in everyday engineering environments.
with today's advanced technology like photoshop, paint etc. we need to understand some basic concepts like how they are cropping the image , tilt the image etc.
In our presentation you will find basic introduction of 2D transformation.
Shortest path search for real road networks and dynamic costs with pgRoutingantonpa
ย
This presentation will show the inside and current state of pgRouting
development. It will explain the shortest path search in real road
networks and how the data structure is important for getting better
routing results. We will show how you can improve the quality of the search with dynamic costs and make the result look closer to the reality. We will demonstrate the way of using pgRouting together with other Open Source tools. Also you will learn about difficulties and limitations of implementing routing functionality in GIS applications, the difference between algorithms and their performance.
pgRouting is an extension of PostgreSQL and PostGIS. A predecessor of
pgRouting - pgDijkstra, written by Sylvain Pasche from Camptocamp, was
extended by Orkney (Japan) and renamed to pgRouting, which now is a part of the PostLBS project.
pgRouting can perform:
* shortest path search (with 3 different algorithms)
* Traveling Salesperson Problem solution (TSP)
* driving distance geometry calculation
This Presentation Is Specially Made For Those Engineering Students Who are In Gujarat Technological University. This Presentation Clears Your All Doubts About Basics Fundamentals of Numerical Integration. Also You Will Learn Different Types Of Error Formula To Solve the Numerical Integration Sum.
Here we have included details about relaxation method and some examples .
Contribution - Parinda Rajapakha, Hashan Wanniarachchi, Sameera Horawalawithana, Thilina Gamalath, Samudra Herath and Pavithri Fernando.
This presentation covers scalar quantity, vector quantity, addition of vectors & multiplication of vector. I hope this PPT will be helpful for Instructors as well as students.
Concepts and Applications of the Fundamental Theorem of Line Integrals.pdfJacobBraginsky
ย
A three-part examination of the Fundamental Theorem of Line Integrals. Learn how to use this theorem in multivariable calculus. Simplify the process of solving line integrals using the FTLI.
A Probabilistic Algorithm for Computation of Polynomial Greatest Common with ...mathsjournal
ย
In the earlier work, Knuth present an algorithm to decrease the coefficient growth in the Euclidean algorithm of polynomials called subresultant algorithm. However, the output polynomials may have a small factor which can be removed. Then later, Brown of Bell Telephone Laboratories showed the subresultant in another way by adding a variant called ๐ and gave a way to compute the variant. Nevertheless, the way failed to determine every ๐ correctly.
In this paper, we will give a probabilistic algorithm to determine the variant ๐ correctly in most cases by adding a few steps instead of computing ๐ก(๐ฅ) when given ๐(๐ฅ) and๐(๐ฅ) โ โค[๐ฅ], where ๐ก(๐ฅ) satisfies that ๐ (๐ฅ)๐(๐ฅ) + ๐ก(๐ฅ)๐(๐ฅ) = ๐(๐ฅ), here ๐ก(๐ฅ), ๐ (๐ฅ) โ โค[๐ฅ]
A Non Local Boundary Value Problem with Integral Boundary ConditionIJMERJOURNAL
ย
ABSTRACT: In this article a three point boundary value problem associated with a second order differential equation with integral type boundary conditions is proposed. Then its solution is developed with the help of the Greenโs function associated with the homogeneous equation. Using this idea and Iteration method is proposed to solve the corresponding linear problem.
https://telecombcn-dl.github.io/2017-dlai/
Deep learning technologies are at the core of the current revolution in artificial intelligence for multimedia data analysis. The convergence of large-scale annotated datasets and affordable GPU hardware has allowed the training of neural networks for data analysis tasks which were previously addressed with hand-crafted features. Architectures such as convolutional neural networks, recurrent neural networks or Q-nets for reinforcement learning have shaped a brand new scenario in signal processing. This course will cover the basic principles of deep learning from both an algorithmic and computational perspectives.
Fixed Point Results for Weakly Compatible Mappings in Convex G-Metric Spaceinventionjournals
ย
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Matrix Transformations on Some Difference Sequence SpacesIOSR Journals
ย
The sequence spaces ๐โ(๐ข,๐ฃ,ฮ), ๐0(๐ข,๐ฃ,ฮ) and ๐(๐ข,๐ฃ,ฮ) were recently introduced. The matrix classes (๐ ๐ข,๐ฃ,ฮ :๐) and (๐ ๐ข,๐ฃ,ฮ :๐โ) were characterized. The object of this paper is to further determine the necessary and sufficient conditions on an infinite matrix to characterize the matrix classes (๐ ๐ข,๐ฃ,ฮ โถ๐๐ ) and (๐ ๐ข,๐ฃ,ฮ โถ ๐๐). It is observed that the later characterizations are additions to the existing ones
Design of Second Order Digital Differentiator and Integrator Using Forward Di...inventionjournals
ย
In this paper, the second order differentiator and integrator, design is investigated. Firstly, the forward difference formula is applied in numerical differentiation for deriving the transfer function of second order differentiator and integrator. Thereafter, the Richardson extrapolation is used for generating the high accuracy results, while using low order formulas. Further, the conventional Lagrange FIR fractional delay filter is applied directly for implementation of the second order differentiator, design. Finally, the effectiveness of this new design approach is illustrated by using several numerical examples.
Dual Spaces of Generalized Cesaro Sequence Space and Related Matrix Mappinginventionjournals
ย
In this paper we define the generalized Cesaro sequence spaces ํํํ (ํ, ํ, ํ ). We prove the space ํํํ (ํ, ํ, ํ ) is a complete paranorm space. In section-2 we determine its Kothe-Toeplitz dual. In section-3 we establish necessary and sufficient conditions for a matrix A to map ํํํ ํ, ํ, ํ to ํโ and ํํํ (ํ, ํ, ํ ) to c, where ํโ is the space of all bounded sequences and c is the space of all convergent sequences. We also get some known and unknown results as remarks.
A presentation slides of Jihoon Ko*, Yunbum Kook* and Kijung Shin, "Incremental Lossless Graph Summarization", KDD 2020.
Given a fully dynamic graph, represented as a stream of edge insertions and deletions, how can we obtain and incrementally update a lossless summary of its current snapshot?
As large-scale graphs are prevalent, concisely representing them is inevitable for efficient storage and analysis. Lossless graph summarization is an effective graph-compression technique with many desirable properties. It aims to compactly represent the input graph as (a) a summary graph consisting of supernodes (i.e., sets of nodes) and superedges (i.e., edges between supernodes), which provide a rough description, and (b) edge corrections which fix errors induced by the rough description. While a number of batch algorithms, suited for static graphs, have been developed for rapid and compact graph summarization, they are highly inefficient in terms of time and space for dynamic graphs, which are common in practice.
In this work, we propose MoSSo, the first incremental algorithm for lossless summarization of fully dynamic graphs. In response to each change in the input graph, MoSSo updates the output representation by repeatedly moving nodes among supernodes. MoSSo decides nodes to be moved and their destinations carefully but rapidly based on several novel ideas. Through extensive experiments on 10 real graphs, we show MoSSo is (a) Fast and 'any time': processing each change in near-constant time (less than 0.1 millisecond), up to 7 orders of magnitude faster than running state-of-the-art batch methods, (b) Scalable: summarizing graphs with hundreds of millions of edges, requiring sub-linear memory during the process, and (c) Effective: achieving comparable compression ratios even to state-of-the-art batch methods.
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologistโs survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
NO1 Uk best vashikaran specialist in delhi vashikaran baba near me online vas...Amil Baba Dawood bangali
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Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
ย
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
Learn about the cost savings, reduced environmental impact, and minimal disruption associated with trenchless technology. Discover detailed explanations of popular techniques such as pipe bursting, cured-in-place pipe (CIPP) lining, and directional drilling. Understand how these methods can be applied to various types of infrastructure, from residential plumbing to large-scale municipal systems.
Ideal for homeowners, contractors, engineers, and anyone interested in modern plumbing solutions, this guide provides valuable insights into why trenchless pipe repair is becoming the preferred choice for pipe rehabilitation. Stay informed about the latest advancements and best practices in the field.
About
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
โข Remote control: Parallel or serial interface.
โข Compatible with MAFI CCR system.
โข Compatible with IDM8000 CCR.
โข Compatible with Backplane mount serial communication.
โข Compatible with commercial and Defence aviation CCR system.
โข Remote control system for accessing CCR and allied system over serial or TCP.
โข Indigenized local Support/presence in India.
โข Easy in configuration using DIP switches.
Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
โข Remote control: Parallel or serial interface
โข Compatible with MAFI CCR system
โข Copatiable with IDM8000 CCR
โข Compatible with Backplane mount serial communication.
โข Compatible with commercial and Defence aviation CCR system.
โข Remote control system for accessing CCR and allied system over serial or TCP.
โข Indigenized local Support/presence in India.
Application
โข Remote control: Parallel or serial interface.
โข Compatible with MAFI CCR system.
โข Compatible with IDM8000 CCR.
โข Compatible with Backplane mount serial communication.
โข Compatible with commercial and Defence aviation CCR system.
โข Remote control system for accessing CCR and allied system over serial or TCP.
โข Indigenized local Support/presence in India.
โข Easy in configuration using DIP switches.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
Sachpazis:Terzaghi Bearing Capacity Estimation in simple terms with Calculati...Dr.Costas Sachpazis
ย
Terzaghi's soil bearing capacity theory, developed by Karl Terzaghi, is a fundamental principle in geotechnical engineering used to determine the bearing capacity of shallow foundations. This theory provides a method to calculate the ultimate bearing capacity of soil, which is the maximum load per unit area that the soil can support without undergoing shear failure. The Calculation HTML Code included.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
ย
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
Cosmetic shop management system project report.pdfKamal Acharya
ย
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
ย
This presentation is about the working procedure of Shahjalal Fertilizer Company Limited (SFCL). A Govt. owned Company of Bangladesh Chemical Industries Corporation under Ministry of Industries.
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
2. โ Embedding Symbolic and Hierarchical Data
โก Introduction to Hyperbolic Space
โข Optimization over Hyperbolic Space
โฃ Toy Experiments
Overview
2
4. Symbolic and Hierarchical Data
4
Symbolic data with Implicit hierarchy.
Downstream tasks
link prediction, node classification, community detection, visualization
Wordnet Twitter Social Graph
?LINK
community
5. Good Hierarchical Embedding
5
For downstream tasks, symbolic and hierarchical data needs to
be embedded into space.
Good Embedding?
Embeddings of similar symbols should aggregate in some sense.
Symbolic arithmetic exists: v(King)- v(man) + v(woman)=v(Queen)
Hierarchy can be restored from embedded data.
The space should have low dimension.
7. Limitation of Euclidean Embedding
7
Embed graph structure while preserving distances
Thm) Trees cannot be embedded into Euclidean space with
arbitrarily low distortion for any number of dimensions
a
b Graph Euclidean ??
D(a,b) 2 0.1 1.889
D(a,c) 2 1 1.902
D(a,d) 2 1.8 1.962
Euclidean
Graph
??
c
d
a
b
c
d
a
b
c
d
Embedding
Representation tradeoffs for hyperbolic Embeddings (ICML 2018)
12. Suggested loss function
12
A Example of loss function over hyperbolic space.
Fundamentally, gradients of loss tells which direction the points
should proceed.
Poincarรฉ Embeddings for Learning Hierarchical Representations (ICML 2017)
13. Gradient Descent Algorithm
13
Input: ๐: ๐ฟ2 โ โ, ๐0 โ ๐ฟ2, ๐ = 0
repeat
choose a descent direction ๐ฃ ๐ โ ๐๐ ๐
๐ฟ2
choose a retraction ๐ ๐ ๐
: ๐๐ ๐
๐ฟ2
โ ๐ฟ2
choose a step length ๐ผ ๐ โ โ
set ๐ ๐+1 = ๐ ๐ ๐
(๐ผ ๐ ๐ฃ ๐)
๐ โ ๐ + 1
until ๐ ๐+1 sufficiently minimize ๐
Nothing different from usual gradient descent except for
Gradient direction
Retraction
Optimization methods on Riemannian manifolds and their application to shape space (SIAM 2012)
14. Gradient Descent Algorithm
14
Input: ๐: ๐ฟ2 โ โ, ๐0 โ ๐ฟ2, ๐ = 0
repeat
choose a descent direction ๐ฃ ๐ โ ๐๐ ๐
๐ฟ2
choose a retraction ๐ ๐ ๐
: ๐๐ ๐
๐ฟ2
โ ๐ฟ2
choose a step length ๐ผ ๐ โ โ
set ๐ ๐+1 = ๐ ๐ ๐
(๐ผ ๐ ๐ฃ ๐)
๐ โ ๐ + 1
until ๐ ๐+1 sufficiently minimize ๐
What is the gradient on Hyperbolic space?
๐ โถ (โ2
, โ๐๐ฅ0 2
+ ๐๐ฅ1 2
+ ๐๐ฅ ๐ 2
) โ โ
โ๐ ?
16. Gradient Descent Algorithm
16
Input: ๐: ๐ฟ2 โ โ, ๐0 โ ๐ฟ2, ๐ = 0
repeat
choose a descent direction ๐ฃ ๐ โ ๐๐ ๐
๐ฟ2
choose a retraction ๐ ๐ ๐
: ๐๐ ๐
๐ฟ2
โ ๐ฟ2
choose a step length ๐ผ ๐ โ โ
set ๐ ๐+1 = ๐ ๐ ๐
(๐ผ ๐ ๐ฃ ๐)
๐ โ ๐ + 1
until ๐ ๐+1 sufficiently minimize ๐
What is the retraction on Hyperbolic space?
17. Hyperboloid model
17
Retraction tells how ends points of tangent vectors correspond
to the point on manifold.
We chose affine geodesic as retraction
๐พ๐ก = cosh ||๐ฃ||โ ๐ก ๐ + sinh ||๐ฃ||โ ๐ก
๐ฃ
||๐ฃ||โ
๐โฒ โ ๐ฟ2
๐ (๐โฒ
) โ ๐ฟ2
At ๐ โ ๐ฟ2 with direction ๐ฃ โ ๐๐ ๐ฟ2
23. Takeaways
23
Hyperbolic space is promising to represent symbolic and
hierarchical datasets.
Geometry determines path toward optimal points.
Regardless of optimization technique, the optimal point is only
depends on loss function.
Interpretation: Can the path entail semantics?
Loss function over hyperbolic space should be discreetly
chosen.
Is it suitable for given geometry? Differentiable? / operation?
Unfortunately, we loose simple arithmetic.
Editor's Notes
Good evening. I am segwang kim from machine intelligence lab. My topic is Hierarchical representation with hyperbolic geometry.
This topic is the topic I am currently working on, but I have gotten nothing meaningful yet. I found this topic intriguing in that it suggests alternative ways to represent symbolic and hierarchical datasets, which in turns helps to do downstream tasks in Natural Language Processing or Social Network Analysis.
This is an overview.
The main goal of this talk is to make you get along with hyperbolic representation.
First, I will introduce the data of interest to be represented and conventional way to embed those datasets.
Second, I will go over shortcomings of conventional embedding and introduce the gist of hyperbolic space.
Third, I am gonna show optimization technique over hyperbolic space.
In the end, Toy Experiments are followed.
Recent papers are included in this presentation.
The datasets I am dealing with, such as wordnet or social network are symbolic and hierarchical. They are symbolic because words or users have no meaningful numeric values. They are just symbols. On top of that, they are hierarchical since there exist partial orderings between data points like dogs belong to mammals and mammals belong to animal. Or, when a twitter user follows another, then we can have ordering between them.
The typical machine learning problem on those datasets are link prediction, node classification, community detection or visualization. To be specific, someone would ask are sprinkler and birdcage linked? or what community does a particular user belong to?
To tackle those problems, we need to parametrize symbolic and hierarchical dataset into numeric forms. We call this process as embedding. Once data points are embedded into some space, we can apply a machine learning model that work on the space.
Even if symbolic datapoints are represented in numerical form, it is natural to expect that the embedding should agree on our intuition.
For instance, two words with similar meaning should be represented as two points that are close to each other.
This two-dimensional figure seems to catch semantic relation.
Like this, we expect some properties from good embedding.
Down the ages, we have embedded symbolic data into the most familiar space, Euclidean space.
However, there are some limitations of Euclidean Embedding.
To illustrate, assume that we want to solve machine learning problem on this bushy-structured datasets. Edge between two nodes means they have something in common.
Therefore, we would want to find the embedding that preserves distances among nodes measured in the graph.
Unfortunately, a second you embed the data points into two dimensional Euclidean space, you would realize that the huge distortions have been made. While the graph distance between node a and b is 2, the Euclidean distance between corresponding points is far less than 2.
To remedy this problem, researchers have increased the dimensionality of Euclidean space. However, by doing that, we loose opportunities to analyze it low dimension.
On top of that, trying to embed trees into Euclidean space is wrong from the beginning.
To be more formally, there is a theorem that Trees cannot be โฆ.
So, main question is, what if we have a space that preserve graph structure well like this one? What is this mysterious space? Now, itโs time to introduce hyperbolic space.
Time for series of math slides.
The best analogy I can use for introducing hyperbolic space is Euclidean space.
We can define geometry of given space or manifold by looking into its domain and inner product structure on tangent space.
Before elaborating why inner product structure does matter, letโs formally define hyperbolic space.
Hyperbolic space is a manifold with constant sectional curvature -1 and five different models are used for describing it. Actually they are same because there exists isometries among them.
Anyhow, I pick one of them. A Poicare disk model.
The domain of N-dimensional poicare disk model is N-dimensional sphere. A innerproduct of tangent space is defined like this.
Unlike Euclidean space which has the same innerprdocut rule for all tangent space, hyperbolic space has different innerproduct structure depending on which point given tangent space is attached. In mathetmatical term, this is called Riemannian metric.
To compare these two spaces, letโs do an inner product.
First you attach tangent plane to given point p in Euclidean or hyperbolic space and then, you pick two arbitrary tangent vectors from the tangent plane. In case of Euclidean product, you take component-wise product and do summation. Note that the point p has nothing to do with computing inner product.
However, in case of hyperbolic space, this highlighted term is multiplied after usual inner product. Note that it depends on point p. Because of this term, strange things are happened.
As I said, inner product of tangent space governs geometry of space. Because, it defines length, angle and โlineโ of given space.
From the calculus 101, we know that length of given path is defined as line integral of norm of instantaneous velocity, which is tangent vector. Since norm is defined when inner product is given, the Riemannian Metric comes into play.
Also, angle between two tangent vectors is governed by innerproduct structure because inner products need to be done.
Finally, if we keep in mind that line is not defined as straight path but the shortest path connecting starting and end points, shape of line in hyperbolic space must be different.
The shortest path is the optimal solution of this functional equation which seems almost impossible to solve.
But Mathematician concludes that line in hyperbolic space is either an usual arc which perpendicularly intersects with boundary of n-dimensional sphere or straight line starting from the center.
Considering the norm of tangent vector increases as base point goes to boundary, the shortest path must be inclined to pass region around center rather than near boundary. So it must be tilted toward center.
One interesting fact about hyperbolic space is we can choose a one model among five ones depending on situation. Fundamentally, they are all same because of existence of isometry.
The paper โ โ suggest that Poincare ball model is more adequate for visualization than Lorentz model, defined like this. This is because Lorentz model is defined on ambient space with constraints. But Lorentz model guarantees more computational stability of gradient than Poincare ball model.
In the following optimization section, I will explain optimization technique on Lorentz model not Poincare model.
This is one example of loss function over hyperbolic space.
As you can see, this loss function has hyperbolic distance terms.
Details are omitted, but basically, this disperses irrelevant datapoints and aggregates relevant ones.
Because gradients of loss tells which direction the datapoints should proceed, we need to know how to compute derivative of given loss function.
This is Riemannian Gradient descent algorithm.
There are only two parts you need to focus on. First, choosing a descent direction, second choosing a retraction
Choosing a descent direction needs more a little bit of efforts than usual gradient.
Letโs assume that we want to minimize a loss function over two-dimensional Lorentz model.
Basically, we want to find gradient of f.
It takes two steps.
Basically, we need to correspond naรฏve gradients to a tangent vector.
First, once we get a gradient from tensorflow or any api, as shown in blue box, this value is unique no matter which metric tensor you have chosen.
If we interpret gradient as linear mapping from tangent space to real number, Riesz representation theorem implies that there is a corresponding vector such that inner product with the vector is the gradient map.
To find the vector, inverse of metric tensor needs to be multiplied to usual derivatives in order to compensates extra terms in hyperbolic innerproduct.
It is complicated but, bottom line is just flip the sign of the first element of usual gradient.
The second step is projection.
Because Lorentz model is defined in ambient space, we need to project the resulting vector from the first step to tangent place of model. It only takes some multiplication and addition.
Therefore, we can get Riemannian descent direction by flipping signs of all components of hyperbolic gradient of the loss.
Retraction tells how can a point be moved to given direction.
When the point is moved to the tip of the direction, it escapes a manifold. This is sad.
However, the point is moved to the tip of the geodesics, then it stays on the manifold and we are happy.
The geodesic is a hyperbolic version of line and this simple formula is all you need.
The last step is trivial. We just need to iterate previous steps until we get sufficiently small errors.