introduction to machine learning cs725 instructor prof
play

Introduction to Machine Learning - CS725 Instructor: Prof. Ganesh - PowerPoint PPT Presentation

Nov 28, 2023 •317 likes •666 views

Introduction to Machine Learning - CS725 Instructor: Prof. Ganesh Ramakrishnan Lecture 13 - KKT Conditions, Duality, SVR Dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  1. Introduction to Machine Learning - CS725 Instructor: Prof. Ganesh Ramakrishnan Lecture 13 - KKT Conditions, Duality, SVR Dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  2. KKT conditions for SVR m L ( w , α, α ∗ , µ, µ ∗ ) = 1 2 ∥ w ∥ 2 + C ( ) ∑ ∑ ( ξ i + ξ ∗ y i − w ⊤ φ ( x i ) − b − ϵ − ξ i i ) + + α i i =1 i m m m ( ) ∑ α ∗ b + w ⊤ φ ( x i ) − y i − ϵ − ξ ∗ ∑ ∑ µ ∗ i ξ ∗ − µ i ξ i − i i i i =1 i =1 i =1 Differentiating the Lagrangian w.r.t. w , m ∑ w − α i φ ( x i ) + α ∗ ( α i − α ∗ i φ ( x i ) = 0 i.e. , w = i ) φ ( x i ) i =1 Differentiating the Lagrangian w.r.t. ξ i , C − α i − µ i = 0 i.e. , α i + µ i = C Differentiating the Lagrangian w.r.t ξ ∗ i , α ∗ i + µ ∗ i = C Differentiating the Lagrangian w.r.t b , i ( α ∗ ∑ i − α i ) = 0 Complimentary slackness: α i ( y i − w ⊤ φ ( x i ) − b − ϵ − ξ i ) = 0 AND µ i ξ i = 0 AND α ∗ i ( b + w ⊤ φ ( x i ) − y i − ϵ − ξ ∗ i ) = 0 AND µ ∗ i ξ ∗ i = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  3. For Support Vector Regression, since the original objective and the constraints are convex, any ( w , b , α, α ∗ , µ, µ ∗ , ξ, ξ ∗ ) that satisfy the necessary KKT conditions gives optimality (conditions are also sufficient) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  4. Some observations α i , α ∗ i ≥ 0, µ i , µ ∗ i ≥ 0, α i + µ i = C and α ∗ i + µ ∗ i = C Thus, α i , µ i , α ∗ i , µ ∗ i ∈ [0 , C ], ∀ i If 0 < α i < C , then 0 < µ i < C (as α i + µ i = C ) µ i ξ i = 0 and α i ( y i − w ⊤ φ ( x i ) − b − ϵ − ξ i ) = 0 are complementary slackness conditions So 0 < α i < C ⇒ ξ i = 0 and y i − w ⊤ φ ( x i ) − b = ϵ + ξ i = ϵ All such points lie on the boundary of the ϵ band Using any point x j (that is with α j ∈ (0 , C )) on margin, we can recover b as: b = y j − w ⊤ φ ( x j ) − ϵ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  5. Support Vector Regression Dual Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  6. Weak Duality L ∗ ( α, α ∗ , µ, µ ∗ ) = w , b ,ξ,ξ ∗ L ( w , b , ξ, ξ ∗ , α, α ∗ , µ, µ ∗ ) min By weak duality theorem, we have: 2 ∥ w ∥ 2 + C ∑ m 1 i =1 ( ξ i + ξ ∗ i ) ≥ L ∗ ( α, α ∗ , µ, µ ∗ ) min w , b ,ξ,ξ ∗ s.t. y i − w ⊤ φ ( x i ) − b ≤ ϵ − ξ i , and w ⊤ φ ( x i ) + b − y i ≤ ϵ − ξ ∗ i , and ξ i , ξ ∗ ≥ 0, ∀ i = 1 , . . . , n The above is true for any α i , α ∗ i ≥ 0 and µ i , µ ∗ i ≥ 0 Thus, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  7. Weak Duality L ∗ ( α, α ∗ , µ, µ ∗ ) = w , b ,ξ,ξ ∗ L ( w , b , ξ, ξ ∗ , α, α ∗ , µ, µ ∗ ) min By weak duality theorem, we have: 2 ∥ w ∥ 2 + C ∑ m 1 i =1 ( ξ i + ξ ∗ i ) ≥ L ∗ ( α, α ∗ , µ, µ ∗ ) min w , b ,ξ,ξ ∗ s.t. y i − w ⊤ φ ( x i ) − b ≤ ϵ − ξ i , and w ⊤ φ ( x i ) + b − y i ≤ ϵ − ξ ∗ i , and ξ i , ξ ∗ ≥ 0, ∀ i = 1 , . . . , n The above is true for any α i , α ∗ i ≥ 0 and µ i , µ ∗ i ≥ 0 Thus, m 1 2 ∥ w ∥ 2 + C ∑ ( ξ i + ξ ∗ α,α ∗ ,µ,µ ∗ L ∗ ( α, α ∗ , µ, µ ∗ ) min i ) ≥ max w , b ,ξ,ξ ∗ i =1 s.t. y i − w ⊤ φ ( x i ) − b ≤ ϵ − ξ i , and w ⊤ φ ( x i ) + b − y i ≤ ϵ − ξ ∗ i , and ξ i , ξ ∗ ≥ 0, ∀ i = 1 , . . . , n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  8. Dual objective L ∗ ( α, α ∗ , µ, µ ∗ ) = w , b ,ξ,ξ ∗ L ( w , b , ξ, ξ ∗ , α, α ∗ , µ, µ ∗ ) min Assume: In case of SVR, we have a strictly convex objective and linear constraints ⇒ KKT conditions are necessary and sufficient and strong duality holds: m 1 2 ∥ w ∥ 2 + C ∑ ( ξ i + ξ ∗ α,α ∗ ,µ,µ ∗ L ∗ ( α, α ∗ , µ, µ ∗ ) min i ) = max w , b ,ξ,ξ ∗ i =1 s.t. y i − w ⊤ φ ( x i ) − b ≤ ϵ − ξ i , and w ⊤ φ ( x i ) + b − y i ≤ ϵ − ξ ∗ i , and ξ i , ξ ∗ ≥ 0, ∀ i = 1 , . . . , n This value is precisely obtained at the ( w , b , ξ, ξ ∗ , α, α ∗ , µ, µ ∗ ) that satisfies the necessary (and sufficient) KKT optimality conditions Given strong duality, we can equivalently solve α,α ∗ ,µ,µ ∗ L ∗ ( α, α ∗ , µ, µ ∗ ) max . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  9. 2 ∥ w ∥ 2 + C ∑ m L ( α, α ∗ , µ, µ ∗ ) = 1 i =1 ( ξ i + ξ ∗ i ) + m α i ( y i − w ⊤ φ ( x i ) − b − ϵ − ξ i ) + α ∗ i ( w ⊤ φ ( x i ) + b − y i − ϵ − ξ ∗ ( ∑ i ) i =1 m ( µ i ξ i + µ ∗ i ξ ∗ ∑ i ) i =1 i in terms of α , α ∗ , µ and µ ∗ by using We obtain w , b , ξ i , ξ ∗ m ( α i − α ∗ the KKT conditions derived earlier as w = ∑ i ) φ ( x i ) i =1 m ( α i − α ∗ i ) = 0 and α i + µ i = C and α ∗ i + µ ∗ and ∑ i = C i =1 Thus, we get: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  11. 2 ∥ w ∥ 2 + C ∑ m L ( α, α ∗ , µ, µ ∗ ) = 1 i =1 ( ξ i + ξ ∗ i ) + m α i ( y i − w ⊤ φ ( x i ) − b − ϵ − ξ i ) + α ∗ i ( w ⊤ φ ( x i ) + b − y i − ϵ − ξ ∗ ( ∑ i ) i =1 m ( µ i ξ i + µ ∗ i ξ ∗ ∑ i ) i =1 i in terms of α , α ∗ , µ and µ ∗ by using We obtain w , b , ξ i , ξ ∗ m ( α i − α ∗ the KKT conditions derived earlier as w = ∑ i ) φ ( x i ) i =1 m ( α i − α ∗ i ) = 0 and α i + µ i = C and α ∗ i + µ ∗ and ∑ i = C i =1 Thus, we get: L ( w , b , ξ, ξ ∗ , α, α ∗ , µ, µ ∗ ) = 1 j ( α i − α ∗ i )( α j − α ∗ j ) φ ⊤ ( x i ) φ ( x j ) + ∑ ∑ 2 i i ( ξ i ( C − α i − µ i ) + ξ ∗ i ( C − α ∗ i − µ ∗ i ( α i − α ∗ ∑ i )) − b ∑ i ) − i ( α i + α ∗ i y i ( α i − α ∗ j ( α i − α ∗ ϵ ∑ i ) + ∑ i ) − ∑ ∑ i )( α j − i α ∗ j ) φ ⊤ ( x i ) φ ( x j ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  12. 2 ∥ w ∥ 2 + C ∑ m L ( α, α ∗ , µ, µ ∗ ) = 1 i =1 ( ξ i + ξ ∗ i ) + m α i ( y i − w ⊤ φ ( x i ) − b − ϵ − ξ i ) + α ∗ i ( w ⊤ φ ( x i ) + b − y i − ϵ − ξ ∗ ( ∑ i ) i =1 m ( µ i ξ i + µ ∗ i ξ ∗ ∑ i ) i =1 i in terms of α , α ∗ , µ and µ ∗ by using We obtain w , b , ξ i , ξ ∗ m ( α i − α ∗ the KKT conditions derived earlier as w = ∑ i ) φ ( x i ) i =1 m ( α i − α ∗ i ) = 0 and α i + µ i = C and α ∗ i + µ ∗ and ∑ i = C i =1 Thus, we get: L ( w , b , ξ, ξ ∗ , α, α ∗ , µ, µ ∗ ) = 1 j ( α i − α ∗ i )( α j − α ∗ j ) φ ⊤ ( x i ) φ ( x j ) + ∑ ∑ 2 i i ( ξ i ( C − α i − µ i ) + ξ ∗ i ( C − α ∗ i − µ ∗ i ( α i − α ∗ ∑ i )) − b ∑ i ) − i ( α i + α ∗ i y i ( α i − α ∗ j ( α i − α ∗ ϵ ∑ i ) + ∑ i ) − ∑ ∑ i )( α j − i α ∗ j ) φ ⊤ ( x i ) φ ( x j ) = − 1 j ( α i − α ∗ i )( α j − α ∗ j ) φ ⊤ ( x i ) φ ( x j ) − ϵ ∑ ∑ ∑ i ( α i + 2 i α ∗ i y i ( α i − α ∗ i ) + ∑ i ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  13. Kernel function: K ( x i , x j ) = φ T ( x i ) φ ( x j ) w = ∑ m i =1 ( α i − α ∗ i ) φ ( x i ) ⇒ the final decision function f ( x ) = w T φ ( x ) + b = ∑ m i =1 ( α i − α ∗ i ) φ T ( x i ) φ ( x )+ y j − ∑ m i =1 ( α i − α ∗ i ) φ T ( x i ) φ ( x j ) − ϵ x j is any point with α j ∈ (0 , C ). Recall similarity with . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  14. Kernel function: K ( x i , x j ) = φ T ( x i ) φ ( x j ) w = ∑ m i =1 ( α i − α ∗ i ) φ ( x i ) ⇒ the final decision function f ( x ) = w T φ ( x ) + b = ∑ m i =1 ( α i − α ∗ i ) φ T ( x i ) φ ( x )+ y j − ∑ m i =1 ( α i − α ∗ i ) φ T ( x i ) φ ( x j ) − ϵ x j is any point with α j ∈ (0 , C ). Recall similarity with kernelized expression for Ridge Regression The dual optimization problem to compute the α ’s for SVR is: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

Introduction to Machine Learning - CS725 Instructor: Prof. Ganesh Ramakrishnan Overview of Linear

Introduction to Machine Learning - CS725 Instructor: Prof. Ganesh Ramakrishnan Overview of Linear

Introduction to Machine Learning - CS725 Instructor: Prof. Ganesh Ramakrishnan Overview of Linear Algebra Solving Linear Equation: Geometric View Simple example of two equations and two unknowns x and y to be found: 2 x y = 0 and x + 2 y

535 views • 41 slides
Introduction to Machine Learning - CS725 Instructor: Prof. Ganesh Ramakrishnan Lecture 7 - Linear

Introduction to Machine Learning - CS725 Instructor: Prof. Ganesh Ramakrishnan Lecture 7 - Linear

Introduction to Machine Learning - CS725 Instructor: Prof. Ganesh Ramakrishnan Lecture 7 - Linear Regression - Bayesian Inference and Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

654 views • 17 slides
Introduction to Machine Learning - CS725 Instructor: Prof. Ganesh Ramakrishnan Lecture 4 - Linear

Introduction to Machine Learning - CS725 Instructor: Prof. Ganesh Ramakrishnan Lecture 4 - Linear

Introduction to Machine Learning - CS725 Instructor: Prof. Ganesh Ramakrishnan Lecture 4 - Linear Regression - Probabilistic Interpretation and Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

642 views • 21 slides
Introduction to Machine Learning Introduction Prof. Andreas Krause Institute for Machine

Introduction to Machine Learning Introduction Prof. Andreas Krause Institute for Machine

Introduction to Machine Learning Introduction Prof. Andreas Krause Institute for Machine Learning (las.ethz.ch) What is Machine Learning I: An example Classify email messages as Spam or Non Spam Classical Approach : manual rules

1.27k views • 59 slides
CS 337: Arti fi cial Intelligence &amp; Machine Learning Instructor: Prof. Ganesh Ramakrishnan

CS 337: Arti fi cial Intelligence &amp; Machine Learning Instructor: Prof. Ganesh Ramakrishnan

CS 337: Arti fi cial Intelligence &amp; Machine Learning Instructor: Prof. Ganesh Ramakrishnan Lecture 8: Regularization, Overfitting, Bias and Variance August 2019 Recap: Regularization for Generalizability Recall: Complex models could lead to

727 views • 34 slides
Welcome to CS 445 Introduction to Machine Learning Instructor: Dr. Kevin Molloy Announcements

Welcome to CS 445 Introduction to Machine Learning Instructor: Dr. Kevin Molloy Announcements

Welcome to CS 445 Introduction to Machine Learning Instructor: Dr. Kevin Molloy Announcements Workstation Configuration should be complete We will be using Jupyter notebooks on Thursday for class PA 0 is due in 1 week to Autolab

421 views • 21 slides
E9 205 Machine Learning for Signal Processing Introduction to Machine Learning of Sensory Signals

E9 205 Machine Learning for Signal Processing Introduction to Machine Learning of Sensory Signals

E9 205 Machine Learning for Signal Processing Introduction to Machine Learning of Sensory Signals 14-08-2017 Instructor - Sriram Ganapathy (sriram@ee.iisc.ernet.in) Teaching Assistant - Aravind Illa (aravindece77@gmail.com). Overview What

644 views • 30 slides
E9 205 Machine Learning for Signal Processing Introduction to Machine Learning of Sensory Signals

E9 205 Machine Learning for Signal Processing Introduction to Machine Learning of Sensory Signals

E9 205 Machine Learning for Signal Processing Introduction to Machine Learning of Sensory Signals 08-08-2018 Instructor - Sriram Ganapathy (sriramg@iisc.ac.in) Teaching Assistant - Akshara Soman (aksharas@iisc.ac.in). Feature Extraction

983 views • 28 slides
E9 205 Machine Learning for Signal Processing Introduction to Machine Learning of Sensory Signals

E9 205 Machine Learning for Signal Processing Introduction to Machine Learning of Sensory Signals

E9 205 Machine Learning for Signal Processing Introduction to Machine Learning of Sensory Signals 12-08-2019 Instructor - Sriram Ganapathy (sriramg@iisc.ac.in) Teaching Assistant - Prachi Singh (prachisingh@iisc.ac.in)

345 views • 15 slides
E9 205 Machine Learning for Signal Processing Introduction to Machine Learning of Sensory Signals

E9 205 Machine Learning for Signal Processing Introduction to Machine Learning of Sensory Signals

E9 205 Machine Learning for Signal Processing Introduction to Machine Learning of Sensory Signals 19-08-2019 Instructor - Sriram Ganapathy (sriramg@iisc.ac.in) Teaching Assistant - Prachi Singh (prachisingh@iisc.ac.in)

239 views • 21 slides
Welcome to CS 445 Introduction to Machine Learning Instructor: Dr. Kevin Molloy Meet and Greet

Welcome to CS 445 Introduction to Machine Learning Instructor: Dr. Kevin Molloy Meet and Greet

Welcome to CS 445 Introduction to Machine Learning Instructor: Dr. Kevin Molloy Meet and Greet Who is this person? Grew up in Newport News. Last 21 years in Northern Virginia PhD in 2015 in computer science with a focus on robotics,

902 views • 41 slides
Introduction to Machine Learning Linear Regression Prof. Andreas Krause Learning and Adaptive

Introduction to Machine Learning Linear Regression Prof. Andreas Krause Learning and Adaptive

Introduction to Machine Learning Linear Regression Prof. Andreas Krause Learning and Adaptive Systems (las.ethz.ch) Basic Supervised Learning Pipeline Training Data Test Data spam ? Learning Predic- ham Model method ?

566 views • 23 slides
Scalable Machine Learning with Apache Spark Introductions Instructor Introduction

Scalable Machine Learning with Apache Spark Introductions Instructor Introduction

Scalable Machine Learning with Apache Spark Introductions Instructor Introduction Student Introductions Name Professional Responsibilities Fun Personal Interest/Fact Expectations for the Course Course Objectives 1

1.74k views • 147 slides
Exam Review Introduction to Machine Learning T-529-ITME Instructor: Dan Lizotte Exam Logistics

Exam Review Introduction to Machine Learning T-529-ITME Instructor: Dan Lizotte Exam Logistics

Exam Review Introduction to Machine Learning T-529-ITME Instructor: Dan Lizotte Exam Logistics When: Tuesday, 15 May 2007 at 9:00am Where: Ofanleiti 131a, 131b Materials/aids: None. No books, no calculators, no laptops. You

252 views • 9 slides
CS 445 Introduction to Machine Learning Features and the KNN Classifier Instructor: Dr. Kevin

CS 445 Introduction to Machine Learning Features and the KNN Classifier Instructor: Dr. Kevin

CS 445 Introduction to Machine Learning Features and the KNN Classifier Instructor: Dr. Kevin Molloy Quick Review of KNN Classifier If it walks like a duck, and quacks like a duck, it probably is a duck. k = 5 k = 1 Distance (dissimilarity)

686 views • 14 slides
CS 445 Introduction to Machine Learning Logistic Regression Instructor: Dr. Kevin Molloy Review

CS 445 Introduction to Machine Learning Logistic Regression Instructor: Dr. Kevin Molloy Review

CS 445 Introduction to Machine Learning Logistic Regression Instructor: Dr. Kevin Molloy Review Linear regression Finding the weights to assign to a polynomial so that the resulting line minimizes the &quot;loss&quot;. ( ! ,

481 views • 11 slides
Two-Player Perfect Information Games: A Brief Survey Tsan-sheng Hsu tshsu@iis.sinica.edu.tw

Two-Player Perfect Information Games: A Brief Survey Tsan-sheng Hsu tshsu@iis.sinica.edu.tw

Two-Player Perfect Information Games: A Brief Survey Tsan-sheng Hsu tshsu@iis.sinica.edu.tw http://www.iis.sinica.edu.tw/~tshsu 1 Abstract Domain: two-player games. Which game characters are predominant when the solution of a game is the

674 views • 44 slides
Geometric Constraints and Variational Approaches to Image Analysis Daniel Martins Antunes 1

Geometric Constraints and Variational Approaches to Image Analysis Daniel Martins Antunes 1

Geometric Constraints and Variational Approaches to Image Analysis Daniel Martins Antunes 1 Supervised by: Jacques-Olivier Lachaud 1 and Hugues Talbot 2 1 LAMA, Universit Savoie Mont Blanc 2 CentraleSuplec, Universit Paris-Saclay Le

1.77k views • 152 slides
Distinguishing Convergence on Two-Taxon and Three-Taxon Networks Jonathan Mitchell Supervisors:

Distinguishing Convergence on Two-Taxon and Three-Taxon Networks Jonathan Mitchell Supervisors:

References Distinguishing Convergence on Two-Taxon and Three-Taxon Networks Jonathan Mitchell Supervisors: Barbara Holland, Jeremy Sumner University of Tasmania November 6, 2014 . . . . . . . . . . . . . . . . . . . . . .

722 views • 39 slides
Neutrino-Driven Turbulent Convection in Stalled Supernova Cores David Radice Collaborators: E.

Neutrino-Driven Turbulent Convection in Stalled Supernova Cores David Radice Collaborators: E.

Neutrino-Driven Turbulent Convection in Stalled Supernova Cores David Radice Collaborators: E. Abdikamalov, S. Couch, R. Haas, C. Ott, E. Schnetter Contents 1.Turbulence in core-collapse supernovae 2.Numerical simulations 3.Conclusions

890 views • 29 slides
Sierd Cloetingh Roland Oberhnsli Alexander Rudloff TOPO-EUROPE TOPO-EUROPE: geoscience of

Sierd Cloetingh Roland Oberhnsli Alexander Rudloff TOPO-EUROPE TOPO-EUROPE: geoscience of

IGCPs 40s anniversary: global partnership in Earth sciences Sierd Cloetingh Roland Oberhnsli Alexander Rudloff TOPO-EUROPE TOPO-EUROPE: geoscience of coupled surface and lithosphere &amp; mantle processes of continental Europe and

165 views • 5 slides
of Epidemiology (LIRE): The Beginning of the End (or The End of the Beginning?) Jeffrey (Jerry)

of Epidemiology (LIRE): The Beginning of the End (or The End of the Beginning?) Jeffrey (Jerry)

Lumbar Imaging with Reporting of Epidemiology (LIRE): The Beginning of the End (or The End of the Beginning?) Jeffrey (Jerry) Jarvik, MD MPH Departments of Radiology, Neurological Surgery, Health Services Comparative Effectiveness, Cost and

1.02k views • 52 slides
The Sound and the Noise of Black Holes a High Energy Physics Colloquium Edgardo Franzin

The Sound and the Noise of Black Holes a High Energy Physics Colloquium Edgardo Franzin

The Sound and the Noise of Black Holes a High Energy Physics Colloquium Edgardo Franzin Cagliari, 4th July 2016 Outline The experimental setup and the discovery Event horizons and light rings in Schwarzschild Black hole

848 views • 31 slides
linear spaces of tilings Richard Kenyon (Brown University) Thursday, May 12, 16 Rectangle tilings

linear spaces of tilings Richard Kenyon (Brown University) Thursday, May 12, 16 Rectangle tilings

linear spaces of tilings Richard Kenyon (Brown University) Thursday, May 12, 16 Rectangle tilings come in linear families (polytopes) y 2 2 y 1 x 2 x 3 x 1 y 1 &lt; y 2 { P x 1 &lt; x 2 , x 3 Thursday, May 12, 16 GUE minors Given such a

578 views • 20 slides

More recommend