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<title>Null Space Projection - Interactive Demo</title>
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</head>
<body>
<div class="container">
<h1>Null Space Projection</h1>
<p class="subtitle">Interactive visualization of how null space constraints preserve capabilities during model modification</p>
<div class="step-indicator">
<div class="step active" data-step="1" onclick="setStep(1)">1</div>
<div class="step" data-step="2" onclick="setStep(2)">2</div>
<div class="step" data-step="3" onclick="setStep(3)">3</div>
</div>
<div class="main-grid">
<!-- Left Panel: Visualization -->
<div class="panel">
<h2>2D Vector Space Visualization</h2>
<div class="canvas-container">
<canvas id="mainCanvas" width="500" height="500"></canvas>
</div>
<div class="legend">
<div class="legend-item">
<div class="legend-color" style="background: #4caf50;"></div>
<span>Preservation Vector (K)</span>
</div>
<div class="legend-item">
<div class="legend-color" style="background: #f858fb;"></div>
<span>Original Update (ΔW)</span>
</div>
<div class="legend-item">
<div class="legend-color" style="background: #00d4ff;"></div>
<span>Projected Update (ΔW')</span>
</div>
<div class="legend-item">
<div class="legend-color" style="background: rgba(76, 175, 80, 0.2);"></div>
<span>Row Space of K</span>
</div>
<div class="legend-item">
<div class="legend-color" style="background: rgba(0, 212, 255, 0.2);"></div>
<span>Null Space of K</span>
</div>
</div>
</div>
<!-- Right Panel: Controls & Explanation -->
<div class="panel">
<h2>Interactive Controls</h2>
<div class="controls">
<div class="control-group">
<label>Preservation Vector K (what we want to preserve)</label>
<div class="slider-row">
<span>K<sub>x</sub></span>
<input type="range" id="kx" min="-100" max="100" value="80">
<span class="value" id="kx-val">0.80</span>
</div>
<div class="slider-row">
<span>K<sub>y</sub></span>
<input type="range" id="ky" min="-100" max="100" value="30">
<span class="value" id="ky-val">0.30</span>
</div>
</div>
<div class="control-group">
<label>Original Update ΔW (modification we want to apply)</label>
<div class="slider-row">
<span>ΔW<sub>x</sub></span>
<input type="range" id="dwx" min="-100" max="100" value="60">
<span class="value" id="dwx-val">0.60</span>
</div>
<div class="slider-row">
<span>ΔW<sub>y</sub></span>
<input type="range" id="dwy" min="-100" max="100" value="70">
<span class="value" id="dwy-val">0.70</span>
</div>
</div>
<button onclick="animateProjection()">Animate Projection</button>
<button class="secondary" onclick="randomize()">Randomize Vectors</button>
</div>
<div class="results-grid">
<div class="result-box">
<div class="label">K · ΔW (before)</div>
<div class="value" id="dot-before">-</div>
</div>
<div class="result-box">
<div class="label">K · ΔW' (after)</div>
<div class="value" id="dot-after">-</div>
</div>
<div class="result-box">
<div class="label">|ΔW'| / |ΔW|</div>
<div class="value" id="magnitude-ratio">-</div>
</div>
</div>
</div>
<!-- Full-width Math Breakdown -->
<div class="math-breakdown-bar">
<!-- Vector Coordinates Row -->
<div class="vectors-row">
<div class="vector-item">
<span class="vector-label k-color">K</span>
<span class="vector-value">[<span id="bar-kx">0.80</span>, <span id="bar-ky">0.30</span>]</span>
</div>
<div class="vector-item">
<span class="vector-label dw-color">ΔW</span>
<span class="vector-value">[<span id="bar-dwx">0.60</span>, <span id="bar-dwy">0.70</span>]</span>
</div>
<div class="vector-item">
<span class="vector-label dwp-color">ΔW'</span>
<span class="vector-value">[<span id="bar-dwpx">-0.16</span>, <span id="bar-dwpy">0.42</span>]</span>
</div>
</div>
<!-- Calculations Row (3 columns) -->
<div class="calculations-row">
<!-- Dot Product -->
<div class="calc-section">
<div class="section-label">Dot Product (K · ΔW)</div>
<div>= (<span class="k-color" id="dot-kx">0.80</span> × <span class="dw-color" id="dot-dwx">0.60</span>) + (<span class="k-color" id="dot-ky">0.30</span> × <span class="dw-color" id="dot-dwy">0.70</span>)</div>
<div>= <span class="calc-intermediate" id="dot-term1">0.48</span> + <span class="calc-intermediate" id="dot-term2">0.21</span></div>
<div>= <span class="calc-result" id="dot-result">0.69</span></div>
</div>
<!-- Projection Formula -->
<div class="calc-section">
<div class="section-label">Projection Formula</div>
<div><span class="dwp-color">ΔW'</span> = <span class="dw-color">ΔW</span> − [(K·ΔW) / |K|²] × <span class="k-color">K</span></div>
<div style="color:#666">|K|² = <span id="k-mag-sq">0.73</span>, scale = <span id="proj-scale">0.95</span></div>
<div>= [<span class="dw-color" id="proj-dwx">0.60</span>, <span class="dw-color" id="proj-dwy">0.70</span>] − [<span class="calc-intermediate" id="proj-subx">0.76</span>, <span class="calc-intermediate" id="proj-suby">0.28</span>]</div>
<div>= <span class="dwp-color">[<span id="math-dwpx">-0.16</span>, <span id="math-dwpy">0.42</span>]</span></div>
</div>
<!-- Verification -->
<div class="calc-section">
<div class="section-label">Verification (K · ΔW')</div>
<div>= (<span class="k-color" id="ver-kx">0.80</span> × <span class="dwp-color" id="ver-dwpx">-0.16</span>) + (<span class="k-color" id="ver-ky">0.30</span> × <span class="dwp-color" id="ver-dwpy">0.42</span>)</div>
<div>= <span class="calc-intermediate" id="ver-term1">-0.128</span> + <span class="calc-intermediate" id="ver-term2">0.126</span></div>
<div>= <span id="ver-result" class="verification-pass">≈ 0 ✓</span></div>
</div>
</div>
</div>
<!-- Step-by-step explanation -->
<div class="panel full-width">
<div class="tabs">
<button class="tab active" onclick="showTab('concept')">Concept</button>
<button class="tab" onclick="showTab('math')">Math</button>
<button class="tab" onclick="showTab('code')">Code</button>
<button class="tab" onclick="showTab('application')">ML Application</button>
</div>
<div id="concept" class="tab-content active">
<div id="step1-content" class="step-content active">
<h3>Step 1: The Problem</h3>
<div class="explanation">
<p><strong>Goal:</strong> We want to modify a model's weights to remove unwanted behavior (like refusal),
but we don't want to break its useful capabilities (like math, coding, reasoning).</p>
<p style="margin-top: 10px;"><strong>The Challenge:</strong> A naive weight modification <span class="math-inline">ΔW</span>
might accidentally affect the outputs for inputs we care about preserving.</p>
<p style="margin-top: 10px;"><strong>Solution:</strong> Project <span class="math-inline">ΔW</span> into the
<span class="highlight">null space</span> of the preservation activations, ensuring the modification
has <em>zero effect</em> on preserved capabilities.</p>
</div>
</div>
<div id="step2-content" class="step-content">
<h3>Step 2: Understanding Null Space</h3>
<div class="explanation">
<p>The <span class="highlight">null space</span> of a matrix <span class="math-inline">K</span> contains
all vectors <span class="math-inline">x</span> where <span class="math-inline">Kx = 0</span>.</p>
<p style="margin-top: 10px;">In 2D visualization:</p>
<ul style="margin-left: 20px; margin-top: 5px;">
<li>The <span style="color: #4caf50;">green line</span> shows the "row space" of K (directions that K responds to)</li>
<li>The <span style="color: #00d4ff;">blue region</span> shows the "null space" (perpendicular to K)</li>
<li>Any vector in the null space, when multiplied by K, gives zero!</li>
</ul>
</div>
</div>
<div id="step3-content" class="step-content">
<h3>Step 3: Projection & Result</h3>
<div class="explanation">
<p>We decompose the original update <span class="math-inline">ΔW</span> into two parts:</p>
<ul style="margin-left: 20px; margin-top: 5px;">
<li><strong>Row space component:</strong> Part that affects preservation inputs (we <em>remove</em> this)</li>
<li><strong>Null space component:</strong> Part with zero effect on preservation (we <em>keep</em> this)</li>
</ul>
<p style="margin-top: 10px;">The projected update <span class="math-inline">ΔW'</span> = <span class="math-inline">ΔW</span> minus its row-space component.</p>
<p style="margin-top: 10px;"><strong>Result:</strong></p>
<ul style="margin-left: 20px; margin-top: 5px;">
<li><span style="color: #4caf50;">Preservation guaranteed:</span> <span class="math-inline">K · ΔW' = 0</span></li>
<li><span style="color: #f858fb;">Some modification lost:</span> <span class="math-inline">|ΔW'| ≤ |ΔW|</span></li>
</ul>
<p style="margin-top: 10px;"><strong>Trade-off:</strong> The more aligned your update is with preservation directions, the more gets removed.
In practice, refusal behavior often lives in directions somewhat orthogonal to general capabilities, so we can remove most of it while preserving capabilities!</p>
</div>
</div>
</div>
<div id="math" class="tab-content">
<h3>Mathematical Formulation</h3>
<div class="explanation">
<p>Given preservation activations <span class="math-inline">K ∈ ℝ<sup>n×d</sup></span> (n samples, d dimensions):</p>
</div>
<div class="math">
<strong>1. Compute SVD of K:</strong><br>
K = UΣV<sup>T</sup>
</div>
<div class="math">
<strong>2. Build null space projector:</strong><br>
P<sub>null</sub> = I - V<sub>r</sub>V<sub>r</sub><sup>T</sup><br>
<span style="color: #888; font-size: 0.9em;">(where V<sub>r</sub> contains the r significant right singular vectors)</span>
</div>
<div class="math">
<strong>3. Project update into null space:</strong><br>
ΔW' = ΔW · P<sub>null</sub>
</div>
<div class="math">
<strong>4. Verify preservation:</strong><br>
K · ΔW' = K · ΔW · (I - VV<sup>T</sup>) = K · ΔW - K · ΔW · VV<sup>T</sup> ≈ 0
</div>
<div class="explanation" style="margin-top: 20px;">
<p><strong>Why it works:</strong> The rows of V span the row space of K. Subtracting <span class="math-inline">VV<sup>T</sup></span>
removes all components in the row space, leaving only the null space.</p>
</div>
</div>
<div id="code" class="tab-content">
<h3>Implementation</h3>
<div class="code-block">
<span class="keyword">import</span> torch
<span class="keyword">def</span> <span class="function">compute_null_space_projector</span>(K):
<span class="string">"""
Compute projector onto null space of K.
Args:
K: Preservation activations [n_samples, d_model]
Returns:
P_null: Projector matrix [d_model, d_model]
"""</span>
<span class="comment"># Compute SVD (no centering - we want exact null space of K)</span>
U, S, Vh = torch.linalg.svd(K, full_matrices=<span class="keyword">False</span>)
<span class="comment"># Use all right singular vectors (rows of Vh) as row space basis</span>
<span class="comment"># V_r columns span the row space of K</span>
V_r = Vh.T <span class="comment"># [d_model, rank] where rank = min(n_samples, d_model)</span>
<span class="comment"># Null space projector: I - V_r @ V_r.T</span>
<span class="comment"># Projects onto orthogonal complement of row space</span>
P_null = torch.eye(K.shape[<span class="number">1</span>], device=K.device, dtype=K.dtype) - V_r @ V_r.T
<span class="keyword">return</span> P_null
<span class="keyword">def</span> <span class="function">apply_null_space_projection</span>(delta_W, P_null):
<span class="string">"""Project weight update into null space."""</span>
<span class="keyword">return</span> delta_W @ P_null
<span class="comment"># ===========================================</span>
<span class="comment"># Runnable example with synthetic data:</span>
<span class="comment"># ===========================================</span>
<span class="comment"># Simulate preservation activations (e.g., from math/coding prompts)</span>
<span class="comment"># Shape: [n_samples, hidden_dim]</span>
n_samples, hidden_dim = <span class="number">50</span>, <span class="number">128</span>
K = torch.randn(n_samples, hidden_dim)
<span class="comment"># Compute the null space projector</span>
P_null = compute_null_space_projector(K)
<span class="comment"># Simulate a weight update we want to apply (e.g., refusal direction)</span>
delta_W = torch.randn(hidden_dim)
<span class="comment"># Project into null space (safe update)</span>
delta_W_safe = apply_null_space_projection(delta_W, P_null)
<span class="comment"># Verify: K @ delta_W_safe should be ~0 for all samples</span>
effect_on_preservation = K @ delta_W_safe
print(<span class="string">f"Max effect on preservation inputs: {effect_on_preservation.abs().max():.2e}"</span>)
<span class="comment"># Output: Max effect on preservation inputs: ~1e-6 (effectively zero!)</span>
</div>
</div>
<div id="application" class="tab-content">
<h3>Application to Model Abliteration</h3>
<div class="explanation">
<p>In the context of removing refusal behavior from language models:</p>
</div>
<table style="width: 100%; margin: 15px 0; border-collapse: collapse;">
<tr style="background: rgba(0,0,0,0.3);">
<th style="padding: 12px; text-align: left; border-bottom: 1px solid #333;">Component</th>
<th style="padding: 12px; text-align: left; border-bottom: 1px solid #333;">In Demo</th>
<th style="padding: 12px; text-align: left; border-bottom: 1px solid #333;">In Abliteration</th>
</tr>
<tr>
<td style="padding: 12px; border-bottom: 1px solid #222;">K</td>
<td style="padding: 12px; border-bottom: 1px solid #222;">Preservation vector</td>
<td style="padding: 12px; border-bottom: 1px solid #222;">Activations from math, coding, reasoning prompts</td>
</tr>
<tr>
<td style="padding: 12px; border-bottom: 1px solid #222;">ΔW</td>
<td style="padding: 12px; border-bottom: 1px solid #222;">Original update</td>
<td style="padding: 12px; border-bottom: 1px solid #222;">Refusal direction projection</td>
</tr>
<tr>
<td style="padding: 12px; border-bottom: 1px solid #222;">ΔW'</td>
<td style="padding: 12px; border-bottom: 1px solid #222;">Projected update</td>
<td style="padding: 12px; border-bottom: 1px solid #222;">Safe refusal removal (preserves capabilities)</td>
</tr>
<tr>
<td style="padding: 12px;">K · ΔW' = 0</td>
<td style="padding: 12px;">Dot product is zero</td>
<td style="padding: 12px;">Math/coding outputs unchanged</td>
</tr>
</table>
<div class="explanation" style="margin-top: 15px;">
<p><strong>Practical considerations:</strong></p>
<ul style="margin-left: 20px; margin-top: 5px;">
<li>Use diverse preservation prompts (35+ covering math, coding, reasoning, etc.)</li>
<li>rank_ratio of 0.95 keeps most capability variance while allowing some modification</li>
<li>Lower rank_ratio = more aggressive preservation (but less refusal removal)</li>
<li>Compute P_null once per layer, reuse for all weight matrices in that layer</li>
</ul>
</div>
</div>
</div>
</div>
</div>
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const canvas = document.getElementById('mainCanvas');
const ctx = canvas.getContext('2d');
const width = canvas.width;
const height = canvas.height;
const centerX = width / 2;
const centerY = height / 2;
const scale = 200;
// State
let animationProgress = 0;
let isAnimating = false;
let currentStep = 1;
// Get slider values
function getK() {
return {
x: parseFloat(document.getElementById('kx').value) / 100,
y: parseFloat(document.getElementById('ky').value) / 100
};
}
function getDW() {
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x: parseFloat(document.getElementById('dwx').value) / 100,
y: parseFloat(document.getElementById('dwy').value) / 100
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function projectToNullSpace(dw, k) {
// Null space of K is perpendicular to K
// P_null = I - k*k^T / |k|^2
const kNorm = normalize(k);
const projection = dot(dw, kNorm);
return {
x: dw.x - projection * kNorm.x,
y: dw.y - projection * kNorm.y
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function drawNullSpaceRegion(k) {
const norm = normalize(k);
const perp = { x: -norm.y, y: norm.x };
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ctx.beginPath();
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ctx.fill();
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ctx.moveTo(p1.x, p1.y);
ctx.lineTo(p2.x, p2.y);
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function drawRowSpaceRegion(k) {
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const p2 = toCanvas({ x: -norm.x * 2, y: -norm.y * 2 });
const p3 = toCanvas({ x: -norm.x * 2 + perp.x * 0.3, y: -norm.y * 2 + perp.y * 0.3 });
const p4 = toCanvas({ x: norm.x * 2 + perp.x * 0.3, y: norm.y * 2 + perp.y * 0.3 });
ctx.moveTo(p1.x, p1.y);
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ctx.beginPath();
ctx.moveTo(p1.x, p1.y);
ctx.lineTo(p2.x, p2.y);
ctx.stroke();
ctx.setLineDash([]);
}
function drawProjectionLine(dw, dwProjected, k) {
const dwCanvas = toCanvas(dw);
const dwProjCanvas = toCanvas(dwProjected);
ctx.strokeStyle = 'rgba(255, 255, 255, 0.3)';
ctx.lineWidth = 1;
ctx.setLineDash([3, 3]);
ctx.beginPath();
ctx.moveTo(dwCanvas.x, dwCanvas.y);
ctx.lineTo(dwProjCanvas.x, dwProjCanvas.y);
ctx.stroke();
ctx.setLineDash([]);
}
function drawLabel(v, text, color, offsetX = 10, offsetY = -10, showCoords = false) {
const pos = toCanvas(v);
ctx.fillStyle = color;
ctx.font = 'bold 14px Segoe UI';
ctx.fillText(text, pos.x + offsetX, pos.y + offsetY);
// Draw coordinates below the label
if (showCoords) {
ctx.font = '11px Consolas, monospace';
ctx.fillStyle = 'rgba(255,255,255,0.7)';
const coordText = `[${v.x.toFixed(2)}, ${v.y.toFixed(2)}]`;
ctx.fillText(coordText, pos.x + offsetX, pos.y + offsetY + 14);
}
}
function draw() {
ctx.clearRect(0, 0, width, height);
const k = getK();
const dw = getDW();
const dwProjected = projectToNullSpace(dw, k);
// Calculate animation progress for step 3 (vector projection phase)
const step3Start = 0.4;
const vectorAnimProgress = currentStep >= 3 && isAnimating
? Math.min(1, Math.max(0, (animationProgress - step3Start) / (1 - step3Start)))
: 0;
// Interpolate for animation (only during step 3 vector projection phase)
const dwAnimated = {
x: dw.x + (dwProjected.x - dw.x) * vectorAnimProgress,
y: dw.y + (dwProjected.y - dw.y) * vectorAnimProgress
};
drawGrid();
// Draw subspace regions based on current step
if (currentStep >= 2) {
drawRowSpaceRegion(k);
drawNullSpaceRegion(k);
}
// Draw projection line
if (currentStep >= 3 && !isAnimating) {
drawProjectionLine(dw, dwProjected, k);
}
// Draw vectors
const origin = { x: 0, y: 0 };
// Preservation vector K (always show with coordinates)
drawVector(origin, k, '#4caf50', 4);
drawLabel(k, 'K', '#4caf50', 10, -10, true);
if (currentStep >= 3 && isAnimating) {
// During animation: show ΔW fading and morphing into ΔW'
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drawVector(origin, dw, '#f858fb', 3);
drawLabel(dw, 'ΔW', '#f858fb', 10, 15, true);
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// Draw the animating vector (transitioning from ΔW to ΔW')
drawVector(origin, dwAnimated, '#00d4ff', 3);
drawLabel(dwAnimated, "→ ΔW'", '#00d4ff', 10, -10, true);
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drawVector(origin, dw, '#f858fb', 3);
drawLabel(dw, 'ΔW', '#f858fb', 10, 15, true);
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if (currentStep >= 3) {
drawVector(origin, dwProjected, '#00d4ff', 3);
drawLabel(dwProjected, "ΔW'", '#00d4ff', 10, -10, true);
}
}
// Update results
updateResults(k, dw, dwProjected);
}
function updateResults(k, dw, dwProjected) {
const dotBefore = dot(k, dw);
const dotAfter = dot(k, dwProjected);
const magRatio = magnitude(dwProjected) / magnitude(dw);
document.getElementById('dot-before').textContent = dotBefore.toFixed(4);
document.getElementById('dot-after').textContent = dotAfter.toFixed(6);
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afterEl.style.color = Math.abs(dotAfter) < 0.0001 ? '#4caf50' : '#f858fb';
// Update the step-by-step math breakdown
updateMathDisplay(k, dw, dwProjected);
}
function updateMathDisplay(k, dw, dwProjected) {
// Vector coordinates bar (full-width)
document.getElementById('bar-kx').textContent = k.x.toFixed(2);
document.getElementById('bar-ky').textContent = k.y.toFixed(2);
document.getElementById('bar-dwx').textContent = dw.x.toFixed(2);
document.getElementById('bar-dwy').textContent = dw.y.toFixed(2);
document.getElementById('bar-dwpx').textContent = dwProjected.x.toFixed(2);
document.getElementById('bar-dwpy').textContent = dwProjected.y.toFixed(2);
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document.getElementById('dot-ky').textContent = k.y.toFixed(2);
document.getElementById('dot-dwy').textContent = dw.y.toFixed(2);
const term1 = k.x * dw.x;
const term2 = k.y * dw.y;
const dotResult = term1 + term2;
document.getElementById('dot-term1').textContent = term1.toFixed(3);
document.getElementById('dot-term2').textContent = term2.toFixed(3);
document.getElementById('dot-result').textContent = dotResult.toFixed(3);
// Projection formula
const kMagSq = k.x * k.x + k.y * k.y;
const scale = kMagSq > 0 ? dotResult / kMagSq : 0;
const subX = scale * k.x;
const subY = scale * k.y;
document.getElementById('k-mag-sq').textContent = kMagSq.toFixed(3);
document.getElementById('proj-scale').textContent = scale.toFixed(3);
document.getElementById('proj-dwx').textContent = dw.x.toFixed(2);
document.getElementById('proj-dwy').textContent = dw.y.toFixed(2);
document.getElementById('proj-subx').textContent = subX.toFixed(3);
document.getElementById('proj-suby').textContent = subY.toFixed(3);
document.getElementById('math-dwpx').textContent = dwProjected.x.toFixed(3);
document.getElementById('math-dwpy').textContent = dwProjected.y.toFixed(3);
// Verification
document.getElementById('ver-kx').textContent = k.x.toFixed(2);
document.getElementById('ver-dwpx').textContent = dwProjected.x.toFixed(3);
document.getElementById('ver-ky').textContent = k.y.toFixed(2);
document.getElementById('ver-dwpy').textContent = dwProjected.y.toFixed(3);
const verTerm1 = k.x * dwProjected.x;
const verTerm2 = k.y * dwProjected.y;
const verResult = verTerm1 + verTerm2;
document.getElementById('ver-term1').textContent = verTerm1.toFixed(4);
document.getElementById('ver-term2').textContent = verTerm2.toFixed(4);
const verEl = document.getElementById('ver-result');
if (Math.abs(verResult) < 0.0001) {
verEl.textContent = `≈ 0 ✓`;
verEl.className = 'verification-pass';
} else {
verEl.textContent = verResult.toFixed(6);
verEl.className = 'verification-fail';
}
}
function animateProjection() {
if (isAnimating) return;
isAnimating = true;
animationProgress = 0;
// Start from step 1
currentStep = 1;
updateStepIndicators();
// Phase timings (in progress units where 1.0 = complete)
const step2Start = 0.2; // Show subspaces at 20%
const step3Start = 0.4; // Start vector projection at 40%
function animate() {
animationProgress += 0.015;
// Progress through steps based on animation progress
if (animationProgress >= step2Start && currentStep < 2) {
currentStep = 2;
updateStepIndicators();
}
if (animationProgress >= step3Start && currentStep < 3) {
currentStep = 3;
updateStepIndicators();
}
if (animationProgress >= 1) {
animationProgress = 1;
isAnimating = false;
currentStep = 3;
updateStepIndicators();
}
draw();
if (isAnimating) {
requestAnimationFrame(animate);
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animate();
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</body>
</html>
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