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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.
197 lines
17 KiB
HTML
197 lines
17 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>高斯唯一性（Sturm-Liouville）— 交互式探索</title>
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<style>
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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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.reading h3{margin:0;font-family:'Spectral',serif;font-weight:600;font-size:17px;}
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.formula{font:500 13.5px 'JetBrains Mono',monospace;background:#0b0e13;border:1px solid var(--line);border-radius:8px;padding:13px 15px;line-height:1.7;}
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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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.navbar a{font:500 12px 'JetBrains Mono',monospace;color:var(--txt-2);text-decoration:none;padding:5px 11px;border-radius:20px;}
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.navbar a:hover{color:var(--linear);}.navbar a.on{background:var(--linear-soft);color:var(--linear);}
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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;}.grid2{height:520px;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 4 · 定理 2</span>
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<h1>高斯唯一性 <em>Sturm-Liouville</em></h1>
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</div>
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<p class="sub">扫描广义正态分布的形状参数 <b style="color:var(--linear)">α</b>，看密度 p(z;α) 的<strong>得分函数 (log p)′</strong> 是否线性——只有 α=2（高斯）时得分线性，线性可识别性才成立。</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">形状参数 α</div>
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<div class="aval"><span id="aTxt">2.00</span></div>
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<div id="aName" style="margin-top:8px;"></div>
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<input id="alpha" type="range" min="0.6" max="6" step="0.05" value="2" aria-label="形状参数 α" title="形状参数 α" />
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<div class="hint">广义正态 p(z;α) ∝ exp(−|z/β|^α)，β 调成单位方差。α=1 拉普拉斯，α=2 高斯，α→∞ 均匀。</div>
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<div class="preset">
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<button data-a="1">α=1 拉普拉斯</button>
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<button data-a="2">α=2 高斯 ✓</button>
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<button data-a="3">α=3 超高斯</button>
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<button data-a="6">α=6 ≈均匀</button>
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</div>
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</div>
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<div class="block">
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<div class="label">得分函数线性度</div>
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<div id="scoreBadge" class="badge"></div>
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<div class="hint">得分 (log p)′ = −(α/βᵅ)·|z|^{α−1}·sign(z)。仅 α=2 时退化为线性的 −z。</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">⏸ 自动扫描 α</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>密度与得分函数</h2><span>α=2 是唯一让得分变直线的点</span></div>
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<div class="grid2">
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<div class="canvas-wrap"><div class="cap">密度 p(z; α)</div><canvas id="density"></canvas></div>
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<div class="canvas-wrap"><div class="cap">得分函数 (log p)′(z) — 直线 = 高斯</div><canvas id="score"></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>充要链条</h3>
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<div class="formula">p 高斯<br> ⟺ 得分 (log p)′ <span class="hi">线性</span><br> ⟺ 第一特征函数仿射<br> ⟺ <span class="hi">线性可识别性成立</span></div>
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<h3 style="margin-top:2px;">线性可识别性 R²(α)</h3>
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<div class="r2box"><canvas id="r2"></canvas></div>
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<div class="verdict win" id="verdict"></div>
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<div class="src">来源：JEPA/math/04_sturm_liouville_uniqueness.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==='gaussian_uniqueness_explainer.html')a.className='on';nav.appendChild(a);});
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let alpha=2, auto=true, dir=1, lastTick=performance.now();
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function lgamma(x){const g=7,c=[0.99999999999980993,676.5203681218851,-1259.1392167224028,771.32342877765313,-176.61502916214059,12.507343278686905,-0.13857109526572012,9.9843695780195716e-6,1.5056327351493116e-7];if(x<0.5)return Math.log(Math.PI/Math.sin(Math.PI*x))-lgamma(1-x);x-=1;let a=c[0];const t=x+g+0.5;for(let i=1;i<g+2;i++)a+=c[i]/(x+i);return 0.5*Math.log(2*Math.PI)+(x+0.5)*Math.log(t)-t+Math.log(a);}
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function beta_(a){return Math.exp(0.5*(lgamma(1/a)-lgamma(3/a)));}// unit variance scale
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function density(z,a){const b=beta_(a);const norm=a/(2*b*Math.exp(lgamma(1/a)));return norm*Math.exp(-Math.pow(Math.abs(z/b),a));}
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function score(z,a){const b=beta_(a);if(Math.abs(z)<1e-6)return 0;return -(a/Math.pow(b,a))*Math.pow(Math.abs(z),a-1)*Math.sign(z);}
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// R²(α): 启发式建模——得分线性度。线性回归 score≈k·z 的 R²
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function r2OfAlpha(a){
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let sx=0,sy=0,sxx=0,sxy=0,syy=0,n=0;
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for(let z=-3;z<=3;z+=0.05){const s=score(z,a);const wgt=density(z,a);sx+=z*wgt;sy+=s*wgt;sxx+=z*z*wgt;sxy+=z*s*wgt;syy+=s*s*wgt;n+=wgt;}
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const mx=sx/n,my=sy/n;const cov=sxy/n-mx*my,vx=sxx/n-mx*mx,vy=syy/n-my*my;
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const r=cov/Math.sqrt(vx*vy+1e-12);return Math.max(0,Math.min(1,r*r));
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}
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const alphaEl=document.getElementById('alpha'),aTxt=document.getElementById('aTxt'),aName=document.getElementById('aName');
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function nameOf(a){if(a<0.9)return['重尾','#e76f51'];if(a<1.2)return['拉普拉斯 (α≈1)','#f4a261'];if(a<1.7)return['亚高斯','#f4a261'];if(a<2.3)return['高斯 (α=2) ✓','#56d364'];if(a<4)return['超高斯','#f4a261'];return['≈ 均匀','#e76f51'];}
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function setAlpha(v,fromUser){alpha=Math.max(0.6,Math.min(6,v));alphaEl.value=alpha;aTxt.textContent=alpha.toFixed(2);alphaEl.style.setProperty('--fill',((alpha-0.6)/(6-0.6)*100).toFixed(1)+'%');const[nm,col]=nameOf(alpha);aName.innerHTML='<span class="badge" style="background:'+col+'22;color:'+col+'">'+nm+'</span>';if(fromUser){auto=false;const b=document.getElementById('autoBtn');b.classList.remove('live');b.textContent='▶ 自动扫描 α';}draw();}
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alphaEl.addEventListener('input',e=>setAlpha(+e.target.value,true));
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document.querySelectorAll('.preset button').forEach(b=>b.addEventListener('click',()=>setAlpha(+b.dataset.a,true)));
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document.getElementById('autoBtn').addEventListener('click',e=>{auto=!auto;e.target.classList.toggle('live',auto);e.target.textContent=auto?'⏸ 自动扫描 α':'▶ 自动扫描 α';lastTick=performance.now();});
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const dC=document.getElementById('density'),dx=dC.getContext('2d');
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const sC=document.getElementById('score'),sx2=sC.getContext('2d');
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const rC=document.getElementById('r2'),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 drawDensity(){const {w,h,dpr}=fit(dC);dx.setTransform(dpr,0,0,dpr,0,0);dx.clearRect(0,0,w,h);
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const padL=20,padR=16,padT=22,padB=22,x0=padL,x1=w-padR,y0=padT,y1=h-padB,lim=3.4;
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const sx=z=>x0+(z+lim)/(2*lim)*(x1-x0);
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let mx=0;for(let z=-lim;z<=lim;z+=0.05)mx=Math.max(mx,density(z,alpha));
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const sy=v=>y1-v/(mx*1.1)*(y1-y0);
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dx.strokeStyle='#1c2530';dx.lineWidth=1;for(let g=-3;g<=3;g++){dx.beginPath();dx.moveTo(sx(g),y0);dx.lineTo(sx(g),y1);dx.stroke();}
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// 高斯参考
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dx.strokeStyle='rgba(86,211,100,.35)';dx.setLineDash([5,4]);dx.beginPath();for(let i=0,z=-lim;i<=200;i++,z=-lim+2*lim*i/200){const py=sy(density(z,2));i?dx.lineTo(sx(z),py):dx.moveTo(sx(z),py);}dx.stroke();dx.setLineDash([]);
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// 当前密度
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const isG=Math.abs(alpha-2)<0.08;dx.strokeStyle=isG?'#56d364':'#4cc9f0';dx.lineWidth=2.5;dx.shadowColor=dx.strokeStyle;dx.shadowBlur=8;dx.beginPath();for(let i=0,z=-lim;i<=240;i++,z=-lim+2*lim*i/240){const py=sy(density(z,alpha));i?dx.lineTo(sx(z),py):dx.moveTo(sx(z),py);}dx.stroke();dx.shadowBlur=0;
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}
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function drawScore(){const {w,h,dpr}=fit(sC);sx2.setTransform(dpr,0,0,dpr,0,0);sx2.clearRect(0,0,w,h);
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const padL=20,padR=16,padT=18,padB=20,x0=padL,x1=w-padR,y0=padT,y1=h-padB,lim=2.8,vlim=4;
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const sx=z=>x0+(z+lim)/(2*lim)*(x1-x0),sy=v=>y1-(Math.max(-vlim,Math.min(vlim,v))+vlim)/(2*vlim)*(y1-y0);
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sx2.strokeStyle='#1c2530';sx2.lineWidth=1;for(let g=-2;g<=2;g++){sx2.beginPath();sx2.moveTo(sx(g),y0);sx2.lineTo(sx(g),y1);sx2.stroke();}
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sx2.strokeStyle='#33404e';sx2.beginPath();sx2.moveTo(x0,sy(0));sx2.lineTo(x1,sy(0));sx2.stroke();
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// 高斯参考 -z（线性）
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sx2.strokeStyle='rgba(86,211,100,.4)';sx2.setLineDash([5,4]);sx2.lineWidth=1.5;sx2.beginPath();sx2.moveTo(sx(-lim),sy(lim));sx2.lineTo(sx(lim),sy(-lim));sx2.stroke();sx2.setLineDash([]);
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sx2.fillStyle='rgba(86,211,100,.7)';sx2.font="11px 'JetBrains Mono',monospace";sx2.fillText('−z (线性)',x1-78,sy(-lim)+14);
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// 当前 score
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const isG=Math.abs(alpha-2)<0.08;sx2.strokeStyle=isG?'#56d364':'#f4a261';sx2.lineWidth=2.5;sx2.shadowColor=sx2.strokeStyle;sx2.shadowBlur=7;sx2.beginPath();let first=true;for(let i=0,z=-lim;i<=240;i++,z=-lim+2*lim*i/240){const v=score(z,alpha);const py=sy(v);if(first){sx2.moveTo(sx(z),py);first=false;}else sx2.lineTo(sx(z),py);}sx2.stroke();sx2.shadowBlur=0;
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}
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function drawR2(){const {w,h,dpr}=fit(rC);rx.setTransform(dpr,0,0,dpr,0,0);rx.clearRect(0,0,w,h);
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const padL=30,padR=14,padT=16,padB=22,x0=padL,x1=w-padR,y0=padT,y1=h-padB,aMin=0.6,aMax=6;
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const sx=a=>x0+(a-aMin)/(aMax-aMin)*(x1-x0),sy=v=>y1-v*(y1-y0);
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rx.strokeStyle='#1c2530';rx.lineWidth=1;rx.font="10px 'JetBrains Mono',monospace";rx.fillStyle='#5c6672';
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[0,0.5,1].forEach(v=>{rx.beginPath();rx.moveTo(x0,sy(v));rx.lineTo(x1,sy(v));rx.stroke();rx.fillText(v.toFixed(1),4,sy(v)+3);});
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[1,2,3,4,5,6].forEach(a=>{rx.fillText(a,sx(a)-3,y1+15);});
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// 曲线
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rx.strokeStyle='#4cc9f0';rx.lineWidth=2;rx.beginPath();for(let i=0;i<=160;i++){const a=aMin+(aMax-aMin)*i/160;const v=r2OfAlpha(a);i?rx.lineTo(sx(a),sy(v)):rx.moveTo(sx(a),sy(v));}rx.stroke();
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// α=2 峰标
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rx.strokeStyle='rgba(86,211,100,.5)';rx.setLineDash([4,4]);rx.beginPath();rx.moveTo(sx(2),y0);rx.lineTo(sx(2),y1);rx.stroke();rx.setLineDash([]);
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rx.fillStyle='#56d364';rx.font="600 10px 'JetBrains Mono',monospace";rx.fillText('α=2 峰值',sx(2)+4,y0+10);
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// 当前 α 标记
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const cv=r2OfAlpha(alpha);rx.fillStyle=Math.abs(alpha-2)<0.08?'#56d364':'#f4a261';rx.beginPath();rx.arc(sx(alpha),sy(cv),4.5,0,7);rx.fill();
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}
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function draw(){
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drawDensity();drawScore();drawR2();
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const r2=r2OfAlpha(alpha), isG=Math.abs(alpha-2)<0.08;
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const sb=document.getElementById('scoreBadge');
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if(isG){sb.style.background='rgba(86,211,100,.18)';sb.style.color='#56d364';sb.textContent='线性 ✓（−z）';}
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else{sb.style.background='var(--nl-soft)';sb.style.color='#f4a261';sb.textContent='非线性 ✗';}
|
||
const v=document.getElementById('verdict');
|
||
if(isG){v.className='verdict win';v.innerHTML='<b>α = 2：得分函数 = −z（线性）</b><br>第一特征函数仿射 → 线性可识别性成立，R² ≈ '+r2.toFixed(3)+'。高斯是<strong>唯一</strong>满足此条件的分布。';}
|
||
else{v.className='verdict lose';v.innerHTML='当前得分函数<b style="color:var(--nl)">非线性</b>（α≠2），第一特征函数不再仿射 → 线性可识别性失败，R² ≈ <b style="color:var(--nl)">'+r2.toFixed(3)+'</b>。把 α 拨回 <b style="color:var(--ok)">2</b> 看 R² 冲上峰值。';}
|
||
}
|
||
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window.addEventListener('resize',draw);
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setAlpha(2,false);draw();requestAnimationFrame(loop);
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</script>
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</body>
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</html> |