NTT-learning/notebooks/butterfly_mechanics/04_fast_inverse_gs/lecture.ipynb

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  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "pedagogy": {
          "role": "meta",
          "difficulty": 1,
          "kind": "orientation",
          "title": "Objectives"
        }
      },
      "source": "## META | difficulty 1 | Objectives\n\nThis bundle makes the inverse flow explicit.\n\nFocus:\n\n- GS as the fast inverse schedule\n- BO input and NO output\n- why the final `n^-1` scaling appears\n- how bit-reversal fits the forward/inverse pair\n"
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "pedagogy": {
          "role": "mandatory",
          "difficulty": 3,
          "kind": "explanation",
          "title": "GS Feels Like The Same Network Seen From The Other End"
        }
      },
      "source": "## MANDATORY | difficulty 3 | GS Feels Like The Same Network Seen From The Other End\n\nThe inverse is not \u201cmysterious undoing\u201d.\nIt is a staged network with the same family resemblance as CT, but with a different direction of arithmetic and a final scaling.\n"
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "pedagogy": {
          "role": "mandatory",
          "difficulty": 3,
          "kind": "demo",
          "title": "Trace The Exact n=4 GS Paper Example"
        }
      },
      "outputs": [],
      "source": "# MANDATORY | difficulty 3 | Trace The Exact n=4 GS Paper Example\n\nfrom IPython.display import display\n\nfrom ntt_learning.toy_ntt import fast_intt_psi_gs_trace\nfrom ntt_learning.visuals import interactive_trace, plot_butterfly_network, plot_trace_overview, plot_vector_comparison\n\nbo_input = [1467, 3471, 2807, 7621]\ntrace = fast_intt_psi_gs_trace(bo_input, 7681, 1925)\n\nprint(\"unscaled NO output:\", trace.raw_output)\nprint(\"scaled NO output:\", trace.scaled_output)\ndisplay(plot_trace_overview(trace, title=\"GS overview for the n=4 paper example\"))\ndisplay(plot_butterfly_network(trace, title=\"Full GS network for the n=4 paper example\"))\ndisplay(\n    plot_vector_comparison(\n        trace.raw_output,\n        trace.scaled_output,\n        left_label=\"unscaled\",\n        right_label=\"scaled\",\n        title=\"Why the final n^-1 scaling matters\",\n    )\n)\ndisplay(interactive_trace(trace, title=\"GS inverse trace\"))\n"
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "pedagogy": {
          "role": "mandatory",
          "difficulty": 3,
          "kind": "demo",
          "title": "See The Bit-Reversal Map Explicitly"
        }
      },
      "outputs": [],
      "source": "# MANDATORY | difficulty 3 | See The Bit-Reversal Map Explicitly\n\nfrom IPython.display import display\n\nfrom ntt_learning.visuals import plot_bit_reversal_mapping\n\ndisplay(plot_bit_reversal_mapping(4, title=\"Bit-reversal for n=4\"))\ndisplay(plot_bit_reversal_mapping(8, title=\"Bit-reversal for n=8\"))\n"
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "pedagogy": {
          "role": "mandatory",
          "difficulty": 3,
          "kind": "explanation",
          "title": "Why Scaling Waits Until The End"
        }
      },
      "source": "## MANDATORY | difficulty 3 | Why Scaling Waits Until The End\n\nEach GS stage avoids local division by `2`.\nThe accumulated effect of those missing local divisions is corrected by the final multiplication with `n^-1`.\n"
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "pedagogy": {
          "role": "mandatory",
          "difficulty": 3,
          "kind": "demo",
          "title": "Full Forward And Inverse Round Trip"
        }
      },
      "outputs": [],
      "source": "# MANDATORY | difficulty 3 | Full Forward And Inverse Round Trip\n\nfrom IPython.display import display\n\nfrom ntt_learning.toy_ntt import fast_intt_psi_gs_trace, fast_ntt_psi_ct_trace\nfrom ntt_learning.visuals import plot_vector_comparison\n\nsignal = [1, 2, 3, 4]\nforward_trace = fast_ntt_psi_ct_trace(signal, 7681, 1925)\ninverse_trace = fast_intt_psi_gs_trace(forward_trace.raw_output, 7681, 1925)\n\nprint(\"forward BO output:\", forward_trace.raw_output)\nprint(\"inverse scaled output:\", inverse_trace.scaled_output)\ndisplay(\n    plot_vector_comparison(\n        signal,\n        inverse_trace.scaled_output,\n        left_label=\"original\",\n        right_label=\"recovered\",\n        title=\"Forward CT followed by inverse GS\",\n    )\n)\n"
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "pedagogy": {
          "role": "mandatory",
          "difficulty": 2,
          "kind": "quiz",
          "title": "Retrieval Check"
        }
      },
      "source": "## MANDATORY | difficulty 2 | Retrieval Check\n\n1. Why does GS want BO input?\n2. Why does the final scaling not disappear?\n3. What would go wrong if you visually compared GS input and CT NO output without respecting the order change?\n"
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "pedagogy": {
          "role": "facultative",
          "difficulty": 4,
          "kind": "exploration",
          "title": "Optional: A Second GS Example"
        }
      },
      "outputs": [],
      "source": "# FACULTATIVE | difficulty 4 | Optional: A Second GS Example\n\nfrom IPython.display import display\n\nfrom ntt_learning.toy_ntt import fast_intt_psi_gs_trace\nfrom ntt_learning.visuals import interactive_trace\n\ntrace = fast_intt_psi_gs_trace([2489, 6478, 7489, 6607], 7681, 1925)\nprint(\"scaled output:\", trace.scaled_output)\ndisplay(interactive_trace(trace, title=\"Second GS trace\"))\n"
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "pedagogy": {
          "role": "meta",
          "difficulty": 1,
          "kind": "handoff",
          "title": "Next Notebook"
        }
      },
      "source": "## META | difficulty 1 | Next Notebook\n\nNext notebook: `lab.ipynb`\n"
    }
  ],
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      "language": "python",
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    "ntt_learning": {
      "title": "Lecture: Fast Inverse GS",
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