289 - Path Smoothing Question

Optimization using Gradient Descent

• Objective: Minimize two terms using gradient descent.

First Objective

• Update the variable y recursively by:

  • Assigning it the previous y value minus a term proportional to the error deviation.

  • Weight this term with a weighting factor, ( \alpha ), set to 0.5.

Second Term Optimization

• Implementation strategy:

  • Retain the old variable y.

  • Move towards the neighboring value ( y_{i+1} ) while stepping away from ( y_{i} ).

• An improved approach combines terms from both directions:

  • Aim to make ( y_{i} ) close to both ( y_{i-1} ) and ( y_{i+1} ).

  • Optimize this combined term, setting ( \beta ) to 0.1.

• Note: Do not apply this optimization for the first or last node in the sequence.

Implementation Guidelines

• Path Description: Given a 5x5 grid, starting from (0,0) to (4,4), the path moves:

  • Right, down, and then right again.

• Function to Implement:

  • The function smooth takes the path, weighting factors, and a tolerance variable.

  • Creates a new path from the old path (deep copy).

  • The nested function smoother should apply the update equations iteratively, excluding the first and last nodes until the total observed change is smaller than the specified tolerance.

Usage Example

• Compute a new path using the smooth function and verify it by uncommenting the print routine that outputs the result.

Visualization

• The original path and modified path must reflect the unchanged positions of the first and last nodes.

• The in-between values should show a smoother transition:

  • Original path shows constant values in initial steps.

  • Smooth path adjusts values to create a more gradual slope, demonstrating the smoothing effect.

Additional Clarifications

• Corrections to Code:

  • Replace the ending coordinate (4,4) with (4,2).

  • Correct signs from negative to positive in the gradient descent implementation for convergence.

• Iterate through each entry of 2D vectors (e.g., individual ( x_i )) for easier implementation.