Binxu
  • Bio
  • Papers
  • Talks
  • Experience
  • Teaching
  • Recent & Upcoming Talks
    • SfN 2025 Nanosymposium
    • IAIFI Summer Workshop Talk
    • Talk at VSS2025 Session: Theory - Artificial neural networks
    • Talk at VSS2025 Symposium Model-optimized stimuli: more than just pretty pictures
    • Spring into Science Retreat Talk
    • Special Seminar at Washington University Neuroscience Department
    • Yale Cognitive & Neural Computation Lab Talk
    • SfN 2024 Nanosymposium
    • NAISYS2024 Conference Talk
    • KITP Deep Learning Program
    • SfN 2023 Nanosymposium
    • PhD Thesis Defense
    • Kempner Institute Invited Lecture
  • Publications
    • Feature accentuation along the encoding axes of IT neurons uncovers hidden differences in model-brain alignment
    • An analytical theory of power law spectral bias in the learning dynamics of diffusion models
    • Archetypal SAE: Adaptive and Stable Dictionary Learning for Concept Extraction in Large Vision Models
    • The Unreasonable Effectiveness of Gaussian Score Approximation for Diffusion Models and its Applications
    • Diverse capability and scaling of diffusion and auto-regressive models when learning abstract rules
    • Do diffusion models generalize on abstract rules for reasoning?
    • Neural Dynamics of Object Manifold Alignment in the Ventral Stream
    • The hidden linear structure in score-based models and its application
    • Understanding Learning Dynamics of Neural Representations via Feature Visualization at Scale
    • Charting the Landscape of Ventral Stream Neural Code on Generative Image Manifolds
    • Diffusion models generate images like painters: an analytical theory of outline first, details later
    • On the Level Sets and Invariance of Neural Tuning Landscapes
    • Tuning landscapes of the ventral stream
    • High-performance Evolutionary Algorithms for Online Neuron Control
    • Factorized convolution models for interpreting neuron-guided images synthesis
    • On the use of Cortical Magnification and Saccades as Biological Proxies for Data Augmentation
    • The Geometry of Deep Generative Image Models and its Applications
    • Visual prototypes in the ventral stream are attuned to complexity and gaze behavior
    • What constitutes understanding of ventral pathway function?
    • A mechanism for synaptic copy between neural circuits
    • Mutual information and information gating in synfire chains
  • Projects
  • Blog
    • ๐ŸŽ‰ Easily create your own simple yet highly customizable blog
    • ๐Ÿง  Sharpen your thinking with a second brain
    • ๐Ÿ“ˆ Communicate your results effectively with the best data visualizations
    • ๐Ÿ‘ฉ๐Ÿผโ€๐Ÿซ Teach academic courses
    • โœ… Manage your projects
  • Projects
    • PyTorch
    • scikit-learn
    • Learn Python
  • Experience
  • Teaching
    • Neuro140/240 - Biological and Artificial Intelligence
    • Neuro 120 - Introduction to Computational Neuroscience
    • Machine Learning from Scratch
    • BIOL 5648 - Coding and Statistical Thinking in the Neurosciences
  • Academic notes
    • Note on Variants of Diffusion Scheduler, DDPM DDIM PNDM
    • Note on Hippo- Recurrent Memory with Optimal Polynomial Projection
    • Note on S4-Efficiently Modeling Long Sequences with Structured State Spaces
    • Note on Equivalent Score Matching Objective
    • Note on EM algorithm and likelihood lower bound
    • Note on Diffusion Generative Models
    • Probabilistic Models of Neural Population
    • Some Famous Matrix Determinant and Inversion Identities
    • Note on Kernel PCA
    • Analytical Techniques for Finding Spectrum of Tridiagonal Toeplitz Matrix
    • PCA of High Dimensional Trajectory
    • Note on Kernel Ridge Regression
    • Note on Hopfield Network
    • Note on Modern Hopfield Network and Transformers
    • Note on Feedback Stablization
    • Note on Bifurcation Normal Form
    • Note on Invariance and Stable Manifold
      • Invariance
      • Stable and Unstable Manifold Theorem
    • Note on Lyapnov Stability Theory
    • Note on Nonlinear Dynamic System
    • Note on Classic CNNs
    • Note on Word2Vec
    • Note on Non-Parametric Regression
    • Note on Neural Tuning and Information
    • Note on Variational Inference (Updating)
    • Note on Bayesian Optimization
    • Note on Gaussian Process
    • Brief Note on Online Correlation Computation
    • Note on MinMax Game
    • Note on Laplacian Operator (Diffusion) in Geometry Processing
    • Note on Cortical Waves
    • Spectral Graph Theory and Segmentation
    • Computations on Hyperbolic Geometry
    • Note on Deep Reinforcement Learning
    • Note on Photometric Reasoning
    • Note on Generative Adversarial Network
    • Note on Computational Photography
    • Note on Hardware Based Computational Photography
    • Note on Epipolar Geometry and Optic Slow
    • Note on Semantic Vision Task
    • Note on Stereo Problem
    • Note on Image Prior-Markov Random Field Modelling
    • Note on Hessian Aware Zeroth Order Optimization Framework
    • Note on Advanced Computer Vision
    • Note on Image Prior-Spatial Relationship Modeling
    • Krylov Subspace, Conjugate Gradient and Lancosz Iteration
    • Comparison of Major Deep Learning Frameworks (Updating)
    • Note on Online Linear Regression Algorithm
    • LBFGS Algorithm
    • Note on Visual Imagery
    • Note on Local Feature Descriptors
    • Note on ShapeCollage and Patch Based Shape Interpretation
    • Note on Categorization and Concepts
    • Some Computation on Sphere
    • Note on CNN Interpretability
    • Note on Forms of Memory
    • Note on Optimization on Manifold
    • Note on Fragment Based Object Recognition
    • Note on Feature Visualization (Updating)
    • Note on Deep Unsupervised Learning
    • Note on Animal Perception
    • Note on Computer Vision
    • Note on Behavioral Study
    • Note on Network Commnunication
    • This Week's Learning about Brain 4 ็œ‹ๅพ—่ง่ฟ˜ๆ˜ฏ็œ‹ไธ่ง, ่ฟ™ๆ˜ฏไธ€ไธช้—ฎ้ข˜โ€”โ€”่ง†่ง‰ๆ„่ฏ†็š„็ฅž็ป็›ธๅ…ณ็‰ฉ (To see or not to see, that's a question)
    • Note on Algorithm for Non-Negative Matrix Factorization
    • Note on Computation by Synaptic Plasticity
    • Note on Selective Attention
    • Constrained CMA-ES Algorithm
    • Note on Belief Propagation Algorithm (Updating)
    • Distance-Matrix-Based
    • Image-Prior-Reading-Note
    • Note on Belief Propagation for Early Vision
    • Note on Generating Random Shapes
    • Note on VC Dimension
    • Note-on-Fitting-LNP
    • Note-on-Geodesic-and-Curvature
    • Note-on-Graph-Cut
    • Note-on-NEAT-CPPN
    • Note-on-Pareto-Front
    • Note-on-Smoothness-of-Function-on-Manifold
    • Note-on-SNR-for-Neuron
    • Note-on-StyleGAN
    • Note-on-Synaptic-Learning-Rules
    • Optimizer-Design-for-BigGAN
    • pandas_table_wrangling
  • Technical notes
    • Julia for Matlab and Python Users (Updating)
    • JavaScript + HTML Programming 101 (Updating)
    • MTurk 101
    • Scene Programming in MonkeyLogic2
    • Compiling C Extensions for PyTorch
    • Debugging StyleGAN2 in PyTorch
    • Publishing and Using Pretrained model through `Torch.hub`
    • GUI programming in Matlab
    • Hybrid Programming Matlab and Python
    • Importing Geometry from Google Map to Blender
    • Learning CMD / Shell Scripting in Windows
    • Show Animation in PDF Through Latex
    • Transporting Data Between Matlab, Python
    • Procedurally Generating Slides from Script
    • Handling and Printing Parameters in matlab
    • Note on Matlab and Python (matplotlib) Plotting system
    • Windows Working Environment Build Up (Updating)
    • Note on Pandas
    • Autograd Mechanism and Hessian Computation in PyTorch
    • Dynamic Visualization in Python
    • Computer Graphics Environment in Matlab
    • Multi-platform Matlab compiling
    • Working with Caffe
    • Using Google Cloud Service for Large Scale Image Labelling
    • How to Parse Anatomical/Physiological Terms in Latin/Greek
    • Linux Working Environment Build Up (Updating)
    • Understanding conda and python package management (Updating)
    • DeepLabCut Trouble Shooting
    • Common Statistical Problems in Neural Data Analysis (Updating)
    • Resources for Zebrafish Study
    • Note on Automatic 3D Instance Segmentation Pipeline
    • Automatic Behavioral Quantification Pipeline
    • How to translate matlab code into python (numpy)
    • Math Intensive Github Blog Construction
    • 3d-printing-tricks
    • Basic-Java-Concepts
    • Build-RL-Environment
    • Caffe-prototxt-format
    • Common-Recurrent-Modules
    • Compiling C Code for Python
    • Connecting Matlab and TF
    • Drawing-Meshes-in-Illustrator
    • Extracting WeChat Database
    • GUI-programming-python
    • Hacking Matlab Deep Learning
    • IO-Video-in-matlab-python
    • Music21_basics
    • NN_Training_Practicality
    • Note-on-Jax
    • Note_on_Optimizer_Development
    • pytest-usage
    • Python-Logging
    • Stanford-CoreNLP-Usage
    • Translating_TF_and_Torch
    • Vector-Figure-Editing-in-Illustrator

On this page

  • Invariance
  • Properties of an Invariance Set
  • Stable and Unstable Manifold Theorem
Backlinks
  • Note on Nonlinear Dynamic System

Note on Invariance and Stable Manifold

Apr 9, 2021 ยท 1 min read

Invariance

Properties of an Invariance Set

Stable and Unstable Manifold Theorem

Last updated on Apr 9, 2021
Math Geometry Dynamic System
Binxu Wang
Authors
Binxu Wang
Research Fellow
← Note on Bifurcation Normal Form May 9, 2021
Note on Lyapnov Stability Theory Apr 9, 2021 →

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