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- Created planning_explainer.html to demonstrate orthogonal invariance and optimal planning with interactive controls and visualizations. - Created spectral_explainer.html to illustrate spectral decomposition and linear identifiability, featuring adjustable spectral weights and real-time correlation calculations.
177 lines
14 KiB
HTML
177 lines
14 KiB
HTML
<!DOCTYPE html>
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<html lang="zh-CN">
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<head>
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<meta charset="UTF-8" />
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<meta name="viewport" content="width=device-width, initial-scale=1.0" />
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<title>正交不变性与最优规划 — 交互式探索</title>
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canvas{display:block;width:100%;height:100%;}
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.reading{grid-area:read;padding:22px;display:flex;flex-direction:column;gap:16px;overflow-y:auto;}
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.formula .hi{color:var(--linear);}.formula .hl{color:var(--nl);}
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.src{font:400 11px 'JetBrains Mono',monospace;color:var(--txt-3);}
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.navbar{position:fixed;bottom:14px;left:50%;transform:translateX(-50%);display:flex;gap:6px;background:rgba(22,27,34,.9);border:1px solid var(--line);border-radius:30px;padding:6px 10px;-webkit-backdrop-filter:blur(8px);backdrop-filter:blur(8px);z-index:20;}
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@media (max-width:1080px){.app{grid-template-columns:1fr;grid-template-rows:repeat(4,auto);grid-template-areas:"head" "ctrl" "stage" "read";overflow:auto;height:auto;}body{overflow:auto;}.twin{height:360px;flex:none;}}
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</style>
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</head>
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<body>
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<div class="app">
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<header>
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<div style="display:flex;flex-direction:column;gap:6px;">
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<span class="kicker">LeJEPA · Topic 6 · 定理 4</span>
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<h1>正交不变性 <em>与最优规划</em></h1>
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</div>
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<p class="sub">学到的潜空间 <b style="color:var(--nl)">ẑ = Qz</b> 只是真实空间 <b style="color:var(--linear)">z</b> 的旋转。旋转角 θ 任意，只要代价函数是 <b>O(n)-不变的</b>，两边规划的总代价就完全相同。</p>
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</header>
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<section class="card controls">
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<div class="block">
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<div class="label">旋转角 θ（Q ∈ O(n)）</div>
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<div class="aval"><span id="thTxt">45</span><small>°</small></div>
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<input id="theta" type="range" min="0" max="180" step="1" value="45" aria-label="旋转角 θ" title="旋转角 θ" />
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<div class="hint">ẑ = Qz，Q 是旋转 θ 的正交矩阵。无论 θ 多少，点云形状不变（保距离、保角度）。</div>
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</div>
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<div class="block">
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<div class="label">代价函数类型</div>
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<div class="toggle">
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<button id="btnInv" class="on">O(n)-不变<br>‖z−g‖²（距离）</button>
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<button id="btnNon">非不变<br>z₁²（坐标）</button>
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</div>
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<div class="hint">距离类代价旋转不变 → 定理4成立；坐标类代价依赖方向 → 不适用。</div>
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</div>
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<div class="block">
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<div class="label">总规划代价对比</div>
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<div class="costcmp">
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<div class="row"><span style="color:var(--linear)">真实空间 J(z)</span><b id="cz" style="color:var(--linear)">—</b></div>
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<div class="row"><span style="color:var(--nl)">潜空间 J(ẑ)</span><b id="czh" style="color:var(--nl)">—</b></div>
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</div>
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</div>
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<div class="block">
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<div class="label">演示</div>
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<div class="play"><button id="autoBtn" class="live">⏸ 自动旋转 θ</button></div>
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</div>
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</section>
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<section class="card stage">
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<div class="ttl"><h2>两个空间中的最优路径（start → goal）</h2></div>
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<div class="twin">
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<div class="canvas-wrap"><div class="cap" style="color:var(--linear)">真实潜空间 z</div><canvas id="left"></canvas></div>
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<div class="canvas-wrap"><div class="cap" style="color:var(--nl)">学到的潜空间 ẑ = Qz</div><canvas id="right"></canvas></div>
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</div>
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</section>
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<section class="card reading">
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<h3>代价等价（核心）</h3>
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<div class="formula">ℓ(<span class="hl">Qz</span>, a) = ℓ(z, a) <span style="color:var(--txt-3)">（O(n)-不变）</span><br><br>J(a; <span class="hl">Qz₀</span>) = J(a; z₀)<br>→ <span class="hi">â* = a* 且 V̂* = V*</span></div>
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<h3 style="margin-top:2px;">为什么足够？</h3>
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<div class="formula">正交变换保持：<br>‖Qz₁−Qz₂‖ = ‖z₁−z₂‖<br>⟨Qz₁,Qz₂⟩ = ⟨z₁,z₂⟩</div>
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<div class="verdict win" id="verdict"></div>
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<div class="src">来源：JEPA/math/06_planning_equivalence.md</div>
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</section>
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</div>
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<nav class="navbar" id="nav"></nav>
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<script>
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const NAV=[["01","Hermite","hermite_explainer.html"],["02","OU·Mehler","ou_mehler_explainer.html"],["03","谱分解","spectral_explainer.html"],["04","高斯唯一","gaussian_uniqueness_explainer.html"],["05","近似界","approx_bound_explainer.html"],["06","规划等价","planning_explainer.html"]];
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const nav=document.getElementById('nav');NAV.forEach(([n,t,f])=>{const a=document.createElement('a');a.href=f;a.textContent=n+' '+t;if(f==='planning_explainer.html')a.className='on';nav.appendChild(a);});
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let theta=45, invariant=true, auto=true, lastTick=performance.now();
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// 真实空间的 start/goal（固定）
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const start=[-1.6,-1.1], goal=[1.7,1.3];
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function rot(p,deg){const r=deg*Math.PI/180,c=Math.cos(r),s=Math.sin(r);return [c*p[0]-s*p[1], s*p[0]+c*p[1]];}
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// O(n)-不变代价：路径长度 ‖goal-start‖（直线）；非不变：沿路径 z₁² 积分（近似端点平方和）
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function pathCost(s,g,inv){
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if(inv){const dx=g[0]-s[0],dy=g[1]-s[1];return Math.sqrt(dx*dx+dy*dy);}
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// 非不变：对 z₁ 维度的惩罚，沿直线采样 z₁² 平均
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let sum=0,n=20;for(let i=0;i<=n;i++){const t=i/n;const x=s[0]+(g[0]-s[0])*t;sum+=x*x;}return sum/(n+1);
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}
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const thEl=document.getElementById('theta'),thTxt=document.getElementById('thTxt');
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function setTheta(v,fromUser){theta=((v%360)+360)%360;if(theta>180)theta=180;thEl.value=theta;thTxt.textContent=Math.round(theta);thEl.style.setProperty('--fill',(theta/180*100).toFixed(1)+'%');if(fromUser){auto=false;const b=document.getElementById('autoBtn');b.classList.remove('live');b.textContent='▶ 自动旋转 θ';}draw();}
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thEl.addEventListener('input',e=>setTheta(+e.target.value,true));
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document.getElementById('btnInv').addEventListener('click',()=>{invariant=true;document.getElementById('btnInv').classList.add('on');document.getElementById('btnNon').classList.remove('on');draw();});
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document.getElementById('btnNon').addEventListener('click',()=>{invariant=false;document.getElementById('btnNon').classList.add('on');document.getElementById('btnInv').classList.remove('on');draw();});
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document.getElementById('autoBtn').addEventListener('click',e=>{auto=!auto;e.target.classList.toggle('live',auto);e.target.textContent=auto?'⏸ 自动旋转 θ':'▶ 自动旋转 θ';lastTick=performance.now();});
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const lC=document.getElementById('left'),lx=lC.getContext('2d');
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const rC=document.getElementById('right'),rx=rC.getContext('2d');
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function fit(cv){const r=cv.getBoundingClientRect(),dpr=devicePixelRatio||1;cv.width=r.width*dpr;cv.height=r.height*dpr;return{w:r.width,h:r.height,dpr};}
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function drawSpace(ctx,cv,s,g,col,showAxis){
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const {w,h,dpr}=fit(cv);ctx.setTransform(dpr,0,0,dpr,0,0);ctx.clearRect(0,0,w,h);
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const pad=34,x0=pad,x1=w-pad,y0=pad,y1=h-pad,lim=2.6;
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const sx=z=>x0+(z+lim)/(2*lim)*(x1-x0),sy=z=>y1-(z+lim)/(2*lim)*(y1-y0);
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ctx.strokeStyle='#1c2530';ctx.lineWidth=1;for(let gd=-2;gd<=2;gd++){ctx.beginPath();ctx.moveTo(sx(gd),y0);ctx.lineTo(sx(gd),y1);ctx.stroke();ctx.beginPath();ctx.moveTo(x0,sy(gd));ctx.lineTo(x1,sy(gd));ctx.stroke();}
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ctx.strokeStyle='#33404e';ctx.beginPath();ctx.moveTo(x0,sy(0));ctx.lineTo(x1,sy(0));ctx.moveTo(sx(0),y0);ctx.lineTo(sx(0),y1);ctx.stroke();
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// 非不变代价的方向场（z₁² 等高线 = 竖条阴影）
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if(!invariant){for(let xx=-2;xx<=2;xx+=0.25){const a=Math.min(.22,xx*xx*0.06);ctx.fillStyle='rgba(244,162,97,'+a+')';ctx.fillRect(sx(xx),y0,(x1-x0)/(2*lim)*0.25,y1-y0);}}
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// 最优路径直线
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ctx.strokeStyle=col;ctx.lineWidth=3;ctx.setLineDash([]);ctx.beginPath();ctx.moveTo(sx(s[0]),sy(s[1]));ctx.lineTo(sx(g[0]),sy(g[1]));ctx.stroke();
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// start/goal
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ctx.fillStyle='#56d364';ctx.beginPath();ctx.arc(sx(s[0]),sy(s[1]),7,0,7);ctx.fill();ctx.fillStyle='#0b0e13';ctx.font="700 10px 'JetBrains Mono'";ctx.textAlign='center';ctx.fillText('S',sx(s[0]),sy(s[1])+3);
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ctx.fillStyle=col;ctx.beginPath();ctx.arc(sx(g[0]),sy(g[1]),7,0,7);ctx.fill();ctx.fillStyle='#0b0e13';ctx.fillText('G',sx(g[0]),sy(g[1])+3);ctx.textAlign='left';
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}
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function draw(){
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const sR=rot(start,theta), gR=rot(goal,theta);
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drawSpace(lx,lC,start,goal,'#4cc9f0',true);
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drawSpace(rx,rC,sR,gR,'#f4a261',true);
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const cz=pathCost(start,goal,invariant), czh=pathCost(sR,gR,invariant);
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document.getElementById('cz').textContent=cz.toFixed(3);
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document.getElementById('czh').textContent=czh.toFixed(3);
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const eq=Math.abs(cz-czh)<1e-3;
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const v=document.getElementById('verdict');
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if(invariant){v.className='verdict win';v.innerHTML='<b>O(n)-不变代价：J(z) = J(ẑ) = '+cz.toFixed(3)+'</b><br>无论旋转角 θ 多少，两空间总代价完全相等 → 最优动作序列 â* = a*。<strong>线性可识别性足以支持最优规划</strong>。';}
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else{v.className='verdict lose';v.innerHTML='非不变代价 z₁²：J(z) = <b style="color:var(--linear)">'+cz.toFixed(3)+'</b> ≠ J(ẑ) = <b style="color:var(--nl)">'+czh.toFixed(3)+'</b>。<br>代价依赖坐标方向 → 旋转后改变 → <b style="color:var(--nl)">定理4 不适用</b>。这类代价超出 O(n)-不变范围。';}
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}
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function loop(now){if(auto){const dt=(now-lastTick)/1000;lastTick=now;let v=theta+dt*30;if(v>180)v=0;setTheta(v,false);}else lastTick=now;requestAnimationFrame(loop);}
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document.addEventListener('keydown',e=>{if(e.code==='Space'){e.preventDefault();document.getElementById('autoBtn').click();}else if(e.key==='ArrowLeft')setTheta(theta-5,true);else if(e.key==='ArrowRight')setTheta(theta+5,true);});
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window.addEventListener('resize',draw);
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setTheta(45,false);draw();requestAnimationFrame(loop);
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</script>
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</body>
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</html>
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