JuliaNotebooks/Can DPP really sample eigenvalues?.ipynb

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  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "9bca66ee",
   "metadata": {},
   "outputs": [],
   "source": [
    "using LinearAlgebra, SpecialFunctions, SpecialPolynomials, Polynomials, Plots, Distributions"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "755511e4",
   "metadata": {},
   "source": [
    "# Sampling GUE Eigenvalues without forming GUE"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "d006a023",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "ϕ (generic function with 1 method)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "########### Hermite KERNEL ################\n",
    "Γ = gamma\n",
    "Hj(j, x) = basis(Hermite, j)(x) # Hermite Polynomial\n",
    "ϕ(j, x) = exp(-x^2/2)*Hj(j, x)/( π^(1/4)*sqrt(Γ(j+1))*2^(j/2) )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "6b4542e7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Kernel_Hermite (generic function with 2 methods)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Kernel_Hermite(j, x) = j*ϕ(j,x)^2-sqrt(j*(j+1))*ϕ(j-1,x)*ϕ(j+1,x)\n",
    "Kernel_Hermite(j, x, y) = x==y ? Kernel_Hermite(j, x) : sqrt(j/2) * (ϕ(j,x)*ϕ(j-1,y) - ϕ(j-1,x) * ϕ(j,y))/(x-y) "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2da9ac53",
   "metadata": {},
   "source": [
    "# Let us now take $n = 4$ and compare (1) Monte-Carlo and (2) DPP"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "4d5e36f4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n = 4"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "86889492",
   "metadata": {},
   "source": [
    "## Append all GUE eigenvalues, trial = 100000"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "0601b01b",
   "metadata": {},
   "outputs": [],
   "source": [
    "n_trial = 10^5\n",
    "r_MC = []\n",
    "for i in 1:n_trial\n",
    "    A = randn(n, n)+im*randn(n, n)\n",
    "    append!(r_MC, eigvals((A+A')/2))\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "357d89ba",
   "metadata": {},
   "outputs": [
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       "  \"/>\n",
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       "<circle clip-path=\"url(#clip692)\" style=\"fill:#009af9; stroke:none; fill-opacity:0\" cx=\"2157.08\" cy=\"1445.31\" r=\"2\"/>\n",
       "<circle clip-path=\"url(#clip692)\" style=\"fill:#009af9; stroke:none; fill-opacity:0\" cx=\"2187.69\" cy=\"1445.72\" r=\"2\"/>\n",
       "<circle clip-path=\"url(#clip692)\" style=\"fill:#009af9; stroke:none; fill-opacity:0\" cx=\"2218.29\" cy=\"1445.51\" r=\"2\"/>\n",
       "<path clip-path=\"url(#clip690)\" d=\"\n",
       "M1987.39 198.898 L2280.54 198.898 L2280.54 95.2176 L1987.39 95.2176  Z\n",
       "  \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n",
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       "  \"/>\n",
       "<path clip-path=\"url(#clip690)\" d=\"\n",
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      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "histogram(r_MC, normalized = true, bins=80)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c5cab10a",
   "metadata": {},
   "source": [
    "# Create (discretized) Hermite kernel  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "541ce571",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-6.0:0.05:6.0"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dx = 0.05\n",
    "x = -6:dx:6"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "0d9d8e80",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "241×241 Matrix{Float64}:\n",
       "  3.90799e-13   5.13629e-13   6.7323e-13   …  -4.69891e-13  -3.57801e-13\n",
       "  5.13629e-13   6.75069e-13   8.84836e-13     -6.17094e-13  -4.69891e-13\n",
       "  6.7323e-13    8.84836e-13   1.15979e-12     -8.08195e-13  -6.15409e-13\n",
       "  8.80019e-13   1.15663e-12   1.51604e-12     -1.05558e-12  -8.03783e-13\n",
       "  1.14719e-12   1.50778e-12   1.97632e-12     -1.3749e-12   -1.04694e-12\n",
       "  1.49138e-12   1.96017e-12   2.5693e-12   …  -1.78589e-12  -1.3599e-12\n",
       "  1.93353e-12   2.54131e-12   3.33105e-12     -2.31336e-12  -1.76156e-12\n",
       "  2.4999e-12    3.28572e-12   4.30681e-12     -2.98834e-12  -2.27555e-12\n",
       "  3.22329e-12   4.23652e-12   5.55311e-12     -3.84961e-12  -2.93139e-12\n",
       "  4.14458e-12   5.44743e-12   7.14037e-12     -4.94536e-12  -3.7658e-12\n",
       "  5.3145e-12    6.98514e-12   9.15601e-12  …  -6.33539e-12  -4.8243e-12\n",
       "  6.79584e-12   8.93219e-12   1.17082e-11     -8.09356e-12  -6.16314e-12\n",
       "  8.66604e-12   1.13904e-11   1.49304e-11     -1.03108e-11  -7.8516e-12\n",
       "  ⋮                                        ⋱                 ⋮\n",
       " -6.16314e-12  -8.09356e-12  -1.05995e-11      8.93219e-12   6.79584e-12\n",
       " -4.8243e-12   -6.33539e-12  -8.29704e-12  …   6.98514e-12   5.3145e-12\n",
       " -3.7658e-12   -4.94536e-12  -6.47664e-12      5.44743e-12   4.14458e-12\n",
       " -2.93139e-12  -3.84961e-12  -5.04161e-12      4.23652e-12   3.22329e-12\n",
       " -2.27555e-12  -2.98834e-12  -3.91368e-12      3.28572e-12   2.4999e-12\n",
       " -1.76156e-12  -2.31336e-12  -3.0297e-12       2.54131e-12   1.93353e-12\n",
       " -1.3599e-12   -1.78589e-12  -2.33891e-12  …   1.96017e-12   1.49138e-12\n",
       " -1.04694e-12  -1.3749e-12   -1.80065e-12      1.50778e-12   1.14719e-12\n",
       " -8.03783e-13  -1.05558e-12  -1.38246e-12      1.15663e-12   8.80019e-13\n",
       " -6.15409e-13  -8.08195e-13  -1.05847e-12      8.84836e-13   6.7323e-13\n",
       " -4.69891e-13  -6.17094e-13  -8.08195e-13      6.75069e-13   5.13629e-13\n",
       " -3.57801e-13  -4.69891e-13  -6.15409e-13  …   5.13629e-13   3.90799e-13"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K = [Kernel_Hermite(n, xi, xj) for xi in x, xj in x] * dx"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "d271fcb8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "randDPP (generic function with 1 method)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function randprojDPP(Y)\n",
    "    n = size(Y, 2)\n",
    "    J = fill(0, n)\n",
    "    for k in 1:n\n",
    "        p = mean(abs.(Y).^2, dims=2)\n",
    "        J[k] = rand(Categorical(p[:]))\n",
    "        Y = ( Y * qr(Y[J[k], :]).Q )[:, 2:end]\n",
    "    end\n",
    "    return sort(J)\n",
    "end\n",
    "function randDPP(Λ, Y)\n",
    "   mask = rand.(Bernoulli.(Λ))\n",
    "   return randprojDPP(Y[:, mask])#, (mask)\n",
    "end "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "de90c79b",
   "metadata": {},
   "source": [
    "# Sample DPP from the kernel 100000 times"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "784b020a",
   "metadata": {},
   "outputs": [],
   "source": [
    "S, V = eigen(K);\n",
    "S[abs.(S).<1e-10] .= 0.0\n",
    "S[abs.(S).>1.0] .= 1.0\n",
    "r_DPP = [];\n",
    "for i in 1:100000\n",
    "    append!(r_DPP, randDPP(S, V))\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "d3ffdba9",
   "metadata": {},
   "outputs": [
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