Lecture on Direction of Steepest Ascent and Descent

Introduction

• Part of the Maths 2 component from the online BSc Program in Data Science and Programming.
• Focus on the direction of steepest ascent and descent based on gradients and directional derivatives.

Basic Concept

• Water flows downhill in the direction where altitude decreases most rapidly.
• Example: Deccan Plateau, Bangalore and Chennai region showing natural flow direction.

Modeling Water Flow

• Mohan Bhargava's Approach:
  • Models hill using a function h(x).
  • Computes derivative h'(x) to find where it's negative, positive, or zero.
  • Negative derivative indicates flow to the right, positive to the left, zero indicates stationary.
• Gita's Improvement:
  • Uses a two-dimensional model h(x, y) as hills are not one-dimensional.
  • Direction can be more complex than just right or left.

Finding Direction of Water Flow

• Steepest slope indicates direction of fastest altitude decrease.
• Use the directional derivative to determine this direction.
• Minimized when direction is opposite to the gradient vector.

Mathematical Explanation

• Function of n Variables:
  • Use gradient f which should be continuous and exist around point a.
  • Directional derivative fu is gradient f at a point dotted with u.
  • fu is minimized when u is in the opposite direction to the gradient.

Key Results

• Minimization and Maximization:
  • Directional derivative minimized when u is opposite to gradient vector.
  • Maximized when u aligns with the gradient vector.
  • No change when u is perpendicular to the gradient vector.

Practical Examples

• Examples:
  • Function f(x, y) = sin(xy), find gradient and steepest descent/ascent directions.
  • Function gradients guide understanding of steepest slope and constant altitude paths.

Application in Machine Learning

• Gradient Descent:
  • Important concept for minimizing functions, such as in machine learning optimization.
  • Ensures continuity of gradient function, key for applying these principles.

Conclusion

• Understanding the gradient function’s continuity is crucial.
• Correctly applying the concepts ensures effective navigation of slopes and optimization in practical applications.