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icml26/fddf30e3-e5ae-4a68-b862-daa6e531883a/appendix_chunks.jsonl

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+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0142", "section": "A. Frequently Used Notations", "page_start": 15, "page_end": 15, "type": "TableGroup", "text": "Table 3 shows the notations frequently used in this paper. Table 3. Frequently used notations. Symbol Description \\mathcal{D} Dataset of vectors d Dimension of vectors n Dataset size \\boldsymbol{q} Query vector K Retrieval size \\boldsymbol{v} Node to be inserted to a graph index N_{\\rm out}(\\boldsymbol{v}) Out-neighbors of v N_{\\rm in}(\\boldsymbol{v}) In-neighbors of \\boldsymbol{v} \\boldsymbol{u} Candidate neighbor of v to be evaluated L Space partition size \\{\\boldsymbol{w_i}\\}_{i=1}^t t neighbors of \\boldsymbol{u} \\{\\boldsymbol{e_i}\\}_{i=1}^t Edges between \\boldsymbol{u} and \\{\\boldsymbol{w_i}\\}_{i=1}^t \\tau_i Threshold w.r.t. w_i in PRT \\delta_i Threshold w.r.t. w_i in PES R_L Result list W Working set R_F, R_T Dual rings of false positives and ejected nodes m Number of projection vectors in each subspace {\\mathcal F} Set of concatenated projection vectors \\{\\boldsymbol{r_i}\\}_{i=1}^{m^L} Individual projection vectors r_i^* \\in \\mathcal{F} Vector having the smallest angle with w_i - u \\cos \\alpha_i Cosine of the angle between w_i - u and v - u \\cos \\beta_i Cosine of the angle between w_i - u and r_i^* \\cos \\theta_i Cosine of the angle between v-u and r_i^* Cand(v) All nodes visited during the insertion of v H(\\boldsymbol{v}) Nodes in \\operatorname{Cand}(\\boldsymbol{v}) rejected by PES", "source": "marker_v2", "marker_block_id": "/page/14/TableGroup/397"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0143", "section": "B. Proof of Theorem 3.1", "page_start": 15, "page_end": 15, "type": "Text", "text": "In the proof, with slight abuse of notation, we treat u as the origin, and directly use v and each w_i to denote v-u and w_i-u , respectively.", "source": "marker_v2", "marker_block_id": "/page/14/Text/5"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0144", "section": "B. Proof of Theorem 3.1", "page_start": 15, "page_end": 15, "type": "Text", "text": "As such, we have w_1, \\ldots, w_t, v \\in \\mathbb{S}^{d-1} , and \\langle w_i, v \\rangle = \\cos \\alpha_i, i = 1, \\ldots, t . Suppose that d is divisible by L. Let w_i = [w_{i1}, \\ldots, w_{iL}]^{\\top} and v = [v_1, \\ldots, v_L]^{\\top} be the equal-dimension partition of w_i and v, respectively. Suppose that in the l-th subspace, we generate m random projection vectors \\{r_{jl}\\}_{j=1}^m with the norm 1/\\sqrt{L} . We use r_{il}^* to denote the nearest projection vector to w_{il} , i.e.,", "source": "marker_v2", "marker_block_id": "/page/14/Text/6"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0145", "section": "B. Proof of Theorem 3.1", "page_start": 15, "page_end": 15, "type": "Equation", "text": "r_{il}^* = \\underset{1 \\le j \\le m}{\\arg\\max} \\langle r_{jl}, w_{il} \\rangle. \\tag{7}", "source": "marker_v2", "marker_block_id": "/page/14/Equation/7"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0146", "section": "B. Proof of Theorem 3.1", "page_start": 15, "page_end": 15, "type": "Text", "text": "Then, we have \\bm{r_i^*} = [\\bm{r_{i1}^*}, \\dots, \\bm{r_{iL}^*}]^\\top \\in \\mathbb{S}^{d-1}. We introduce", "source": "marker_v2", "marker_block_id": "/page/14/Text/8"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0147", "section": "B. Proof of Theorem 3.1", "page_start": 15, "page_end": 15, "type": "Equation", "text": "C_{il} := \\frac{\\langle \\boldsymbol{w_{il}}, \\boldsymbol{r_{il}^*} \\rangle \\sqrt{L}}{\\|\\boldsymbol{w_{il}}\\|} \\in [-1, 1]. \\tag{8}", "source": "marker_v2", "marker_block_id": "/page/14/Equation/9"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0148", "section": "B. Proof of Theorem 3.1", "page_start": 15, "page_end": 15, "type": "Text", "text": "Let \\mathbb{E}[C_{il}] = \\mu_m and \\mathrm{Var}(C_{il}) = \\sigma_m^2 , where \\mu_m and \\sigma_m^2 depend only on m. Note that they implicitly depend on the subspace dimension d/L. When m grows at a sufficiently high rate compared to d/L, \\mu_m \\to 1 and \\sigma_m \\to 0 . Moreover,", "source": "marker_v2", "marker_block_id": "/page/14/Text/10"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0149", "section": "B. Proof of Theorem 3.1", "page_start": 15, "page_end": 15, "type": "Text", "text": "we have", "source": "marker_v2", "marker_block_id": "/page/14/Text/11"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0150", "section": "B. Proof of Theorem 3.1", "page_start": 15, "page_end": 15, "type": "Equation", "text": "Cov(C_{il}, C_{jl}) = \\rho_m(\\phi_{ijl})\\sigma_m^2 \\tag{9}", "source": "marker_v2", "marker_block_id": "/page/14/Equation/12"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0151", "section": "B. Proof of Theorem 3.1", "page_start": 15, "page_end": 15, "type": "Text", "text": "where \\phi_{ijl} denotes the angle between w_{il} and w_{jl} , and \\rho_m := [0,\\pi] \\to [-1,1] denotes the correlation coefficient function depending on \\phi_{ijl} . Then, we have", "source": "marker_v2", "marker_block_id": "/page/14/Text/13"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0152", "section": "B. Proof of Theorem 3.1", "page_start": 15, "page_end": 15, "type": "Equation", "text": "Y_i := \\Sigma_{l=1}^L \\langle \\boldsymbol{r_{il}^*}, \\boldsymbol{w_{il}} \\rangle = \\Sigma_{l=1}^L \\frac{\\|\\boldsymbol{w_{il}}\\|}{\\sqrt{L}} C_{il}. (10)", "source": "marker_v2", "marker_block_id": "/page/14/Equation/14"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0153", "section": "B. Proof of Theorem 3.1", "page_start": 15, "page_end": 15, "type": "Text", "text": "By the assumption that \\|\\boldsymbol{w_{il}}\\| = (1 + o(1))/\\sqrt{L} , we have", "source": "marker_v2", "marker_block_id": "/page/14/Text/15"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0154", "section": "B. Proof of Theorem 3.1", "page_start": 15, "page_end": 15, "type": "Equation", "text": "\\mathbb{E}[Y_i] = \\mu_m(1 + o(1)). \\tag{11}", "source": "marker_v2", "marker_block_id": "/page/14/Equation/16"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0155", "section": "B. Proof of Theorem 3.1", "page_start": 15, "page_end": 15, "type": "Equation", "text": "Var(Y_i) = \\frac{\\sigma_m^2}{L}(1 + o(1)). (12)", "source": "marker_v2", "marker_block_id": "/page/14/Equation/17"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0156", "section": "B. Proof of Theorem 3.1", "page_start": 15, "page_end": 15, "type": "Equation", "text": "Cov(Y_i, Y_j) = \\frac{\\sigma_m^2}{L^2} \\sum_{l=1}^{L} \\rho_m(\\phi_{ijl}) (1 + o(1)). (13)", "source": "marker_v2", "marker_block_id": "/page/14/Equation/18"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0157", "section": "B. Proof of Theorem 3.1", "page_start": 15, "page_end": 15, "type": "Text", "text": "Then, v_l can be decomposed as v_l = v_{il}^{\\parallel} + v_{il}^{\\perp} , where v_{il}^{\\parallel} and v_{il}^{\\perp} are defined as follows.", "source": "marker_v2", "marker_block_id": "/page/14/Text/19"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0158", "section": "B. Proof of Theorem 3.1", "page_start": 15, "page_end": 15, "type": "Equation", "text": "v_{il}^{\\parallel} = \\frac{\\langle v_l, w_{il} \\rangle}{\\|w_{il}\\|^2} w_{il}, \\quad v_{il}^{\\perp} \\perp w_{il}. (14)", "source": "marker_v2", "marker_block_id": "/page/14/Equation/20"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0159", "section": "B. Proof of Theorem 3.1", "page_start": 15, "page_end": 15, "type": "Text", "text": "Then, we can define Z_i as follows.", "source": "marker_v2", "marker_block_id": "/page/14/Text/21"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0160", "section": "B. Proof of Theorem 3.1", "page_start": 15, "page_end": 15, "type": "Equation", "text": "Z_i := \\langle r_i^*, v \\rangle = Z_i^{(1)} + Z_i^{(2)} (15)", "source": "marker_v2", "marker_block_id": "/page/14/Equation/22"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0161", "section": "B. Proof of Theorem 3.1", "page_start": 15, "page_end": 15, "type": "Text", "text": "where Z_i^{(1)} and Z_i^{(2)} are defined as follows.", "source": "marker_v2", "marker_block_id": "/page/14/Text/23"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0162", "section": "B. Proof of Theorem 3.1", "page_start": 15, "page_end": 15, "type": "Equation", "text": "Z_i^{(1)} := \\sum_{l=1}^{L} \\frac{\\langle \\boldsymbol{v_l}, \\boldsymbol{w_{il}} \\rangle}{\\|\\boldsymbol{w_{il}}\\| \\sqrt{L}} C_{il}. (16)", "source": "marker_v2", "marker_block_id": "/page/14/Equation/24"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0163", "section": "B. Proof of Theorem 3.1", "page_start": 15, "page_end": 15, "type": "Equation", "text": "Z_i^{(2)} := \\sum_{l=1}^L \\langle r_{il}^*, v_{il}^{\\perp} \\rangle. \\tag{17}", "source": "marker_v2", "marker_block_id": "/page/14/Equation/25"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0164", "section": "B. Proof of Theorem 3.1", "page_start": 15, "page_end": 15, "type": "Equation", "text": "\\mathbb{E}[Z_i^{(1)}] = \\frac{\\mu_m}{\\sqrt{L}} \\sum_{l=1}^L \\frac{\\langle \\boldsymbol{v}_l, \\boldsymbol{w}_{il} \\rangle}{\\|\\boldsymbol{w}_{il}\\|} = \\mu_m \\cos \\alpha_i + o(1). (18)", "source": "marker_v2", "marker_block_id": "/page/14/Equation/26"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0165", "section": "B. Proof of Theorem 3.1", "page_start": 15, "page_end": 15, "type": "Text", "text": "By symmetry, we have \\mathbb{E}[Z_i^{(2)}|\\{C_{il}\\}_{l=1}^L]=0 . Then, by the independence of subspaces, we have", "source": "marker_v2", "marker_block_id": "/page/14/Text/27"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0166", "section": "B. Proof of Theorem 3.1", "page_start": 15, "page_end": 15, "type": "Equation", "text": "\\mathbb{E}[Z_i] = \\mu_m \\cos(\\alpha_i) + o(1). \\tag{19}", "source": "marker_v2", "marker_block_id": "/page/14/Equation/28"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0167", "section": "B. Proof of Theorem 3.1", "page_start": 15, "page_end": 15, "type": "Equation", "text": "Var(Z_i^{(1)}) = \\frac{\\sigma_m^2}{L} \\sum_{l=1}^{L} \\frac{\\langle v_l, w_{il} \\rangle^2}{\\|w_{il}\\|^2}. (20)", "source": "marker_v2", "marker_block_id": "/page/14/Equation/29"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0168", "section": "B. Proof of Theorem 3.1", "page_start": 15, "page_end": 15, "type": "Text", "text": "We take the orthogonal decomposition of r_{il}^* as r_{il}^* = r_{il}^{\\parallel} + r_{il}^{\\perp} , where r_{il}^{\\parallel} has the same direction with w_{il} . Let r_{il}^{\\perp} = \\|r_{il}^{\\perp}\\|\\zeta , where \\|\\zeta\\| = 1 is a random vector in a (d/L - 1)-dimensional subspace. We have", "source": "marker_v2", "marker_block_id": "/page/14/Text/30"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0169", "section": "B. Proof of Theorem 3.1", "page_start": 15, "page_end": 15, "type": "Equation", "text": "\\mathbb{E}_{\\zeta}[\\langle \\zeta, v_{il}^{\\perp} \\rangle^{2}] = \\frac{\\|v_{il}^{\\perp}\\|^{2} L}{d - L}. (21)", "source": "marker_v2", "marker_block_id": "/page/14/Equation/31"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0170", "section": "B. Proof of Theorem 3.1", "page_start": 16, "page_end": 16, "type": "Text", "text": "By symmetry, \\mathbb{E}[\\langle r_{il}^{\\perp}, v_{il}^{\\perp} \\rangle \\mid C_{il}] = 0 . Then, we have", "source": "marker_v2", "marker_block_id": "/page/15/Text/1"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0171", "section": "B. Proof of Theorem 3.1", "page_start": 16, "page_end": 16, "type": "Equation", "text": "\\operatorname{Var}(\\langle \\boldsymbol{r}_{il}^*, \\boldsymbol{v}_{il}^{\\perp} \\rangle) = \\mathbb{E}_{C_{il}} \\left[ \\mathbb{E}[\\langle \\boldsymbol{r}_{il}^{\\perp}, \\boldsymbol{v}_{il}^{\\perp} \\rangle^2 \\mid C_{il}] \\right]. \\tag{22}", "source": "marker_v2", "marker_block_id": "/page/15/Equation/2"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0172", "section": "B. Proof of Theorem 3.1", "page_start": 16, "page_end": 16, "type": "Text", "text": "Using \\mathbb{E}[\\|\\boldsymbol{r}_{il}^{\\perp}\\|^2] = \\frac{1}{L}(1 - \\mathbb{E}[C_{il}^2]) , we have", "source": "marker_v2", "marker_block_id": "/page/15/Text/3"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0173", "section": "B. Proof of Theorem 3.1", "page_start": 16, "page_end": 16, "type": "Equation", "text": "Var(Z_i^{(2)}) = \\frac{1 - (\\sigma_m^2 + \\mu_m^2)}{d - L} \\sum_{l=1}^{L} \\| v_{il}^{\\perp} \\|^2. (23)", "source": "marker_v2", "marker_block_id": "/page/15/Equation/4"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0174", "section": "B. Proof of Theorem 3.1", "page_start": 16, "page_end": 16, "type": "Text", "text": "Because Cov(Z_i^{(1)}, Z_i^{(2)}) = 0 , we have", "source": "marker_v2", "marker_block_id": "/page/15/Text/5"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0175", "section": "B. Proof of Theorem 3.1", "page_start": 16, "page_end": 16, "type": "Equation", "text": "Var(Z_i) = Var(Z_i^{(1)}) + Var(Z_i^{(2)}). (24)", "source": "marker_v2", "marker_block_id": "/page/15/Equation/6"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0176", "section": "B. Proof of Theorem 3.1", "page_start": 16, "page_end": 16, "type": "Text", "text": "We define \\epsilon_m:=\\sqrt{1-\\mathbb{E}[C_{il}^2]} . From Eq. (23), we can see that \\mathrm{Var}(Z_i^{(2)})=O(\\epsilon_m^2/L) .", "source": "marker_v2", "marker_block_id": "/page/15/Text/7"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0177", "section": "B. Proof of Theorem 3.1", "page_start": 16, "page_end": 16, "type": "Text", "text": "We now analyze the covariance structure. We cannot assume Z^{(1)} and Z^{(2)} are uncorrelated across different indices i \\neq j . Instead, we decompose the covariance matrix as", "source": "marker_v2", "marker_block_id": "/page/15/Text/8"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0178", "section": "B. Proof of Theorem 3.1", "page_start": 16, "page_end": 16, "type": "Equation", "text": "Cov(Z_i, Z_j) = Cov(Z_i^{(1)}, Z_j^{(1)}) + R_{ij} (25)", "source": "marker_v2", "marker_block_id": "/page/15/Equation/9"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0179", "section": "B. Proof of Theorem 3.1", "page_start": 16, "page_end": 16, "type": "Text", "text": "where the residual term R_{ij} contains the noise auto-covariance and cross-terms:", "source": "marker_v2", "marker_block_id": "/page/15/Text/10"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0180", "section": "B. Proof of Theorem 3.1", "page_start": 16, "page_end": 16, "type": "Equation", "text": "R_{ij} = \\text{Cov}(Z_i^{(2)}, Z_j^{(2)}) + \\text{Cov}(Z_i^{(1)}, Z_j^{(2)}) + \\text{Cov}(Z_i^{(2)}, Z_j^{(1)}). (26)", "source": "marker_v2", "marker_block_id": "/page/15/Equation/11"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0181", "section": "B. Proof of Theorem 3.1", "page_start": 16, "page_end": 16, "type": "Text", "text": "Because {\\rm Var}(Z^{(1)})=O(1/L) and {\\rm Var}(Z^{(2)})=O(\\epsilon_m^2/L) , by the Cauchy-Schwarz inequality, we have", "source": "marker_v2", "marker_block_id": "/page/15/Text/12"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0182", "section": "B. Proof of Theorem 3.1", "page_start": 16, "page_end": 16, "type": "Equation", "text": "|\\text{Cov}(Z_i^{(1)}, Z_j^{(2)})| \\le \\sqrt{\\text{Var}(Z_i^{(1)})\\text{Var}(Z_j^{(2)})} = O\\left(\\frac{\\epsilon_m}{L}\\right).", "source": "marker_v2", "marker_block_id": "/page/15/Equation/13"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0183", "section": "B. Proof of Theorem 3.1", "page_start": 16, "page_end": 16, "type": "Text", "text": "Thus, the entire residual term satisfies R_{ij} = O(\\epsilon_m/L) .", "source": "marker_v2", "marker_block_id": "/page/15/Text/14"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0184", "section": "B. Proof of Theorem 3.1", "page_start": 16, "page_end": 16, "type": "Text", "text": "Now, let us turn to \\mathrm{Cov}(Y_i,Z_j) . When i=j, by conditional expectation, \\mathrm{Cov}(Y_i,Z_j^{(2)})=0 . When i\\neq j , if m\\to\\infty , \\sigma_m^2+\\mu_m^2\\to 1 and \\mathrm{Var}(Z_i^{(2)})\\to 0 . Thus, \\mathrm{Cov}(Y_i,Z_j^{(2)})\\to 0 . Then, as m\\to\\infty , we have", "source": "marker_v2", "marker_block_id": "/page/15/Text/15"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0185", "section": "B. Proof of Theorem 3.1", "page_start": 16, "page_end": 16, "type": "Equation", "text": "Cov(Y_i, Z_j) \\to \\frac{\\sigma_m^2}{L} \\sum_{l=1}^{L} \\rho_m(\\phi_{ijl}) \\langle \\boldsymbol{v_l}, \\boldsymbol{w_{jl}} \\rangle (1 + o(1)). (28)", "source": "marker_v2", "marker_block_id": "/page/15/Equation/16"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0186", "section": "B. Proof of Theorem 3.1", "page_start": 16, "page_end": 16, "type": "Text", "text": "Based on the above analysis, we consider", "source": "marker_v2", "marker_block_id": "/page/15/Text/17"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0187", "section": "B. Proof of Theorem 3.1", "page_start": 16, "page_end": 16, "type": "Equation", "text": "\\boldsymbol{\\xi_{l}} = \\left[ \\frac{\\|\\boldsymbol{w_{1l}}\\|}{\\sqrt{L}} C_{1l}, \\dots, \\frac{\\|\\boldsymbol{w_{tl}}\\|}{\\sqrt{L}} C_{tl}, \\langle \\boldsymbol{r_{1l}^*}, \\boldsymbol{v_l} \\rangle, \\dots, \\langle \\boldsymbol{r_{tl}^*}, \\boldsymbol{v_l} \\rangle \\right]. (29)", "source": "marker_v2", "marker_block_id": "/page/15/Equation/18"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0188", "section": "B. Proof of Theorem 3.1", "page_start": 16, "page_end": 16, "type": "Text", "text": "By the assumptions that \\|v_{il}\\| and \\|w_{il}\\| equal (1 + o(1))/\\sqrt{L} , it can be seen that \\xi_l satisfies the Lyapunov condition. By the independence of difference subspaces, we use the Lindeberg-Feller CLT and obtain the following result.", "source": "marker_v2", "marker_block_id": "/page/15/Text/19"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0189", "section": "B. Proof of Theorem 3.1", "page_start": 16, "page_end": 16, "type": "Equation", "text": "\\sqrt{L}([Y_1, \\dots, Y_t, Z_1, \\dots, Z_t]^{\\top} - \\bar{\\boldsymbol{\\mu}}) \\xrightarrow[L \\to \\infty]{d} \\mathcal{N}(0, \\bar{\\Sigma}_{m,L}) (30)", "source": "marker_v2", "marker_block_id": "/page/15/Equation/20"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0190", "section": "B. Proof of Theorem 3.1", "page_start": 16, "page_end": 16, "type": "Text", "text": "where \\bar{\\mu} = [\\mu_m, \\dots, \\mu_m, \\mu_m \\cos \\alpha_1, \\dots, \\mu_m \\cos \\alpha_t]^\\top , and \\bar{\\Sigma} is represented by block sub-matrices", "source": "marker_v2", "marker_block_id": "/page/15/Text/21"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0191", "section": "B. Proof of Theorem 3.1", "page_start": 16, "page_end": 16, "type": "Equation", "text": "\\bar{\\Sigma} = \\begin{pmatrix} \\Sigma_{YY} & \\Sigma_{YZ} \\\\ \\Sigma_{ZY} & \\Sigma_{ZZ} \\end{pmatrix} \\tag{31}", "source": "marker_v2", "marker_block_id": "/page/15/Equation/22"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0192", "section": "B. Proof of Theorem 3.1", "page_start": 16, "page_end": 16, "type": "Text", "text": "where the elements of \\Sigma_{YY} , \\Sigma_{YZ} and \\Sigma_{ZZ} are L times the covariance computed in Eqs. (13), (28), and (25), respectively.", "source": "marker_v2", "marker_block_id": "/page/15/Text/23"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0193", "section": "B. Proof of Theorem 3.1", "page_start": 16, "page_end": 16, "type": "Text", "text": "Based on this result, Z | Y is still a multivariate normal distribution. We have", "source": "marker_v2", "marker_block_id": "/page/15/Text/24"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0194", "section": "B. Proof of Theorem 3.1", "page_start": 16, "page_end": 16, "type": "Equation", "text": "\\mathbb{E}[\\mathbf{Z}|\\mathbf{Y}] = \\boldsymbol{\\mu}_Z + \\boldsymbol{\\Sigma}_{ZY} \\boldsymbol{\\Sigma}_{YY}^{-1} (\\mathbf{Y} - \\boldsymbol{\\mu}_Y). \\tag{32}", "source": "marker_v2", "marker_block_id": "/page/15/Equation/25"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0195", "section": "B. Proof of Theorem 3.1", "page_start": 16, "page_end": 16, "type": "Equation", "text": "Cov(\\mathbf{Z}|\\mathbf{Y}) = (\\Sigma_{ZZ} - \\Sigma_{ZY}\\Sigma_{YY}^{-1}\\Sigma_{YZ})/L. (33)", "source": "marker_v2", "marker_block_id": "/page/15/Equation/26"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0196", "section": "B. Proof of Theorem 3.1", "page_start": 16, "page_end": 16, "type": "Text", "text": "Let D be the diagonal matrix defined as follows", "source": "marker_v2", "marker_block_id": "/page/15/Text/27"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0197", "section": "B. Proof of Theorem 3.1", "page_start": 16, "page_end": 16, "type": "Equation", "text": "\\mathbf{D} = \\operatorname{diag}[\\cos \\alpha_1, \\dots, \\cos \\alpha_t]. \\tag{34}", "source": "marker_v2", "marker_block_id": "/page/15/Equation/28"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0198", "section": "B. Proof of Theorem 3.1", "page_start": 16, "page_end": 16, "type": "Text", "text": "Because \\Sigma_{ZY} \\to \\mathbf{D}\\Sigma_{YY} , when \\mathbf{Y} = [\\cos \\beta_1, \\dots, \\cos \\beta_t]^\\top , we have", "source": "marker_v2", "marker_block_id": "/page/15/Text/29"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0199", "section": "B. Proof of Theorem 3.1", "page_start": 16, "page_end": 16, "type": "Equation", "text": "\\mathbb{E}[\\mathbf{Z}|\\mathbf{Y}] \\to [\\cos \\alpha_1 \\cos \\beta_1, \\dots, \\cos \\alpha_t \\cos \\beta_t]^\\top. (35)", "source": "marker_v2", "marker_block_id": "/page/15/Equation/30"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0200", "section": "B. Proof of Theorem 3.1", "page_start": 16, "page_end": 16, "type": "Equation", "text": "Cov(\\mathbf{Z}|\\mathbf{Y}) \\to (\\mathbf{\\Sigma}_{\\mathbf{Z}\\mathbf{Z}} - \\mathbf{D}\\mathbf{\\Sigma}_{\\mathbf{Y}\\mathbf{Y}}\\mathbf{D})/L = \\mathbf{R}/L (36)", "source": "marker_v2", "marker_block_id": "/page/15/Equation/31"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0201", "section": "B. Proof of Theorem 3.1", "page_start": 16, "page_end": 16, "type": "Text", "text": "where \\mathbf{R} represents the scaled residual matrix with entries L \\cdot R_{ij} = O(\\epsilon_m) . Then, we conclude.", "source": "marker_v2", "marker_block_id": "/page/15/Text/32"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0202", "section": "B. Proof of Theorem 3.1", "page_start": 16, "page_end": 16, "type": "Text", "text": "Remarks. For each fixed i, if we apply a random rotation matrix to w_i - u and v - u in \\mathbb{R}^d , by the spherical concentration inequality, we can derive the following inequalities.", "source": "marker_v2", "marker_block_id": "/page/15/Text/33"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0203", "section": "B. Proof of Theorem 3.1", "page_start": 16, "page_end": 16, "type": "Equation", "text": "P\\left(\\left|\\|(\\boldsymbol{w_i} - \\boldsymbol{u})_l\\|^2 - \\frac{1}{L}\\right| \\ge \\tilde{\\epsilon}\\right) \\le 2e^{-c_1 d\\tilde{\\epsilon}^2} (37)", "source": "marker_v2", "marker_block_id": "/page/15/Equation/34"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0204", "section": "B. Proof of Theorem 3.1", "page_start": 16, "page_end": 16, "type": "Equation", "text": "P\\left(\\left|\\langle (\\boldsymbol{v}-\\boldsymbol{u})_l, (\\boldsymbol{w_i}-\\boldsymbol{u})_l \\rangle - \\frac{\\cos \\alpha_i}{L}\\right| \\ge \\tilde{\\epsilon}\\right) \\le 2e^{-c_2 d\\tilde{\\epsilon}^2} (38)", "source": "marker_v2", "marker_block_id": "/page/15/Equation/35"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0205", "section": "B. Proof of Theorem 3.1", "page_start": 16, "page_end": 16, "type": "Text", "text": "where c_1 and c_2 are constants. As L \\to \\infty and d/(L^2) \\to \\infty , we have the following results:", "source": "marker_v2", "marker_block_id": "/page/15/Text/36"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0206", "section": "B. Proof of Theorem 3.1", "page_start": 16, "page_end": 16, "type": "Equation", "text": "\\|(\\boldsymbol{w_i} - \\boldsymbol{u})_l\\| = (1 + o_p(1))/\\sqrt{L}. (39)", "source": "marker_v2", "marker_block_id": "/page/15/Equation/37"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0207", "section": "B. Proof of Theorem 3.1", "page_start": 16, "page_end": 16, "type": "Equation", "text": "\\langle (\\boldsymbol{v} - \\boldsymbol{u})_l, (\\boldsymbol{w_i} - \\boldsymbol{u})_l \\rangle = \\frac{\\cos \\alpha}{L} + o_p(\\frac{1}{L}). (40)", "source": "marker_v2", "marker_block_id": "/page/15/Equation/38"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0208", "section": "B. Proof of Theorem 3.1", "page_start": 16, "page_end": 16, "type": "Text", "text": "This implies that L=\\sqrt{d} is a balanced choice, which is consistent with the setting in our experiments.", "source": "marker_v2", "marker_block_id": "/page/15/Text/39"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0209", "section": "C. Related Work", "page_start": 16, "page_end": 16, "type": "Text", "text": "We first supplement Sec. 1.1 with more discussions, and then introduce the preliminaries on probabilistic routing.", "source": "marker_v2", "marker_block_id": "/page/15/Text/41"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0210", "section": "C.1. Discussions on ANNS Solvers", "page_start": 17, "page_end": 17, "type": "Text", "text": "911912", "source": "marker_v2", "marker_block_id": "/page/16/Text/48"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0211", "section": "C.1.1. VECTOR QUANTIZATION", "page_start": 17, "page_end": 17, "type": "Text", "text": "Quantization-based methods for ANNS have a long history. Representative approaches can be found in (Jégou et al., 2011; Ge et al., 2014; Babenko & Lempitsky, 2016; Guo et al., 2020a; Gao & Long, 2024; Gao et al., 2025). Here, we briefly introduce the basic idea of the most widely-used method, Product Quantization (PQ) (Jégou et al., 2011).", "source": "marker_v2", "marker_block_id": "/page/16/Text/3"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0212", "section": "C.1.1. VECTOR QUANTIZATION", "page_start": 17, "page_end": 17, "type": "Text", "text": "Let \\mathcal{D} \\subset \\mathbb{R}^d be a d-dimensional dataset and let q \\in \\mathbb{R}^d be a query vector. PQ divides each data vector x \\in \\mathcal{D} into L sub-vectors, i.e., x = (x_1, x_2, \\ldots, x_L) , where each sub-vector x_j \\in \\mathbb{R}^{d'} has dimension d' = d/L. In this way, PQ constructs L subspaces of dimension d', each containing the corresponding sub-vectors of all data points.", "source": "marker_v2", "marker_block_id": "/page/16/Text/4"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0213", "section": "C.1.1. VECTOR QUANTIZATION", "page_start": 17, "page_end": 17, "type": "Text", "text": "In each subspace, every sub-vector is assigned to one of 2^k quantized sub-vectors. Equivalently, each sub-vector is represented by a sub-codeword of length k bits. The set of all sub-codewords in the j-th subspace is referred to as a codebook and is denoted by \\mathcal{C}_j . By concatenating the sub-codewords across all subspaces, each original vector is represented by a codeword of length kL, taking values in the Cartesian product \\mathcal{C}_1 \\times \\cdots \\times \\mathcal{C}_L .", "source": "marker_v2", "marker_block_id": "/page/16/Text/5"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0214", "section": "C.1.1. VECTOR QUANTIZATION", "page_start": 17, "page_end": 17, "type": "Text", "text": "During the query phase, PQ first computes the distances between the query \\boldsymbol{q} and all quantized sub-vectors in each subspace. For each data vector \\boldsymbol{x} , PQ then sums the corresponding L subspace distances to obtain an approximate distance between \\boldsymbol{x} and \\boldsymbol{q} . After computing the approximate distances for all data vectors, PQ ranks them accordingly and returns the most promising candidates.", "source": "marker_v2", "marker_block_id": "/page/16/Text/6"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0215", "section": "C.1.2. SIMILARITY GRAPH", "page_start": 17, "page_end": 17, "type": "Text", "text": "The workflow of graph-based methods is roughly as follows. In the indexing phase, a similarity graph is constructed as the index structure, where data points serve as nodes and edges connect pairs of nearby nodes. A search can move directly from one node to another only if an edge exists between them. In the query phase, the node with the highest priority in a priority queue is selected, and all of its connected neighbors are visited. During this process, the priority queue is updated whenever a node closer to the query is discovered. The search ends when all nodes in the priority queue have been visited. Finally, the top-K points in the priority queue are returned.", "source": "marker_v2", "marker_block_id": "/page/16/Text/8"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0216", "section": "C.1.2. SIMILARITY GRAPH", "page_start": 17, "page_end": 17, "type": "Text", "text": "Among various graph-based methods, HNSW (Malkov & Yashunin, 2020) is a notable one that employs a multi-layer hierarchical structure to achieve rapid routing. NSG (Fu et al., 2019) optimizes the graph topology to ensure the existence of monotonic paths towards a central entry point, thereby enhancing search efficiency. Vamana, which was introduced along with DiskANN (Subramanya et al., 2019), iteratively refines a random graph into a high-performance graph struc-", "source": "marker_v2", "marker_block_id": "/page/16/Text/9"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0217", "section": "C.1.2. SIMILARITY GRAPH", "page_start": 17, "page_end": 17, "type": "Text", "text": "ture. Despite their structural differences, these methods converge on a similar edge selection criterion, namely RobustPrune , which prioritizes directional diversity over simple proximity to ensure efficient navigation through high-dimensional space.", "source": "marker_v2", "marker_block_id": "/page/16/Text/10"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0218", "section": "C.1.2. SIMILARITY GRAPH", "page_start": 17, "page_end": 17, "type": "Text", "text": "In addition to the three graphs mentioned above, many graph-based methods have been proposed recently (Lu et al., 2021; Gao & Long, 2023; Xie et al., 2025; Wang et al., 2025b). Despite different designs, most of them rely on existing graphs, such as HNSW.", "source": "marker_v2", "marker_block_id": "/page/16/Text/11"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0219", "section": "C.1.3. QUANTIZED GRAPH", "page_start": 17, "page_end": 17, "type": "Text", "text": "As stated in the main body of this paper, QG can achieve high performance on some datasets for small values of K, and in certain cases can be 4 \\times -10 \\times faster than HNSW. However, such improvement is often not robust and suffers from several limitations: (1) Sensitivity to data distribution. The effectiveness of QG relies on the assumption that the ranking induced by quantized distances is sufficiently close to the ranking under the original distances. Unfortunately, this assumption often does not hold, especially for many modern real-world datasets where semantic embeddings are increasingly complex. (2) Sensitivity to K. As K increases, the performance of QG degrades significantly. When K reaches the thousand scale, QG generally has no significant advantage over HNSW. (3) Very large space cost. To improve the accuracy of QG, more bits are required to represent the quantized vectors. As a result, the memory consumption of QG is typically at least 2 \\times larger than HNSW, compromising the use of QG for large datasets.", "source": "marker_v2", "marker_block_id": "/page/16/Text/13"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0220", "section": "C.2.1. RANDOM PROJECTION FOR ANGLE ESTIMATION", "page_start": 17, "page_end": 17, "type": "Text", "text": "Because the norms can be pre-computed, estimating the \\ell_2 distance is equivalent to estimating the cosine of the angle between vectors. In high-dimensional Euclidean spaces, angle estimation via random-projection techniques has been extensively studied, with Locality Sensitive Hashing (LSH) (Indyk & Motwani, 1998; Andoni & Indyk, 2005; 2008) being one of the most influential approaches. Among various LSH methods, SimHash (Charikar, 2002) is a representative one. Its core idea is to generate multiple random hyperplanes that partition the space into cells, so that vectors falling into the same cell are likely to form a small angle with each other. Subsequent studies have proposed more refined strategies for angular distance estimation. In particular, Andoni et al. (2015) introduced Falconn, an LSH method that identifies the projection vector yielding the largest or smallest projection value for a given data vector and uses the corresponding projection index as the hash value. This design leads to substantially improved search performance compared to SimHash. Building on this idea, Pham (2021) further incorporated Con-", "source": "marker_v2", "marker_block_id": "/page/16/Text/16"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0221", "section": "C.2.1. RANDOM PROJECTION FOR ANGLE ESTIMATION", "page_start": 18, "page_end": 18, "type": "Text", "text": "938939", "source": "marker_v2", "marker_block_id": "/page/17/Text/20"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0222", "section": "C.2.1. RANDOM PROJECTION FOR ANGLE ESTIMATION", "page_start": 18, "page_end": 18, "type": "Text", "text": "944945946", "source": "marker_v2", "marker_block_id": "/page/17/Text/24"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0223", "section": "C.2.1. RANDOM PROJECTION FOR ANGLE ESTIMATION", "page_start": 18, "page_end": 18, "type": "Text", "text": "949950", "source": "marker_v2", "marker_block_id": "/page/17/Text/27"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0224", "section": "C.2.1. RANDOM PROJECTION FOR ANGLE ESTIMATION", "page_start": 18, "page_end": 18, "type": "Caption", "text": "Figure 11. DataCompDr data (sampled 50,000 vectors) and query (1,000 vectors) distributions.", "source": "marker_v2", "marker_block_id": "/page/17/Caption/2"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0225", "section": "C.2.1. RANDOM PROJECTION FOR ANGLE ESTIMATION", "page_start": 18, "page_end": 18, "type": "Text", "text": "comitants of Extreme Order Statistics (CEOs) to explicitly identify the projection achieving the maximum or minimum inner product with the data vector, and to record the associated extreme projected value. By exploiting this additional information beyond a discrete hash value, a more accurate estimation of angular distance can be achieved (Pham & Liu, 2022).", "source": "marker_v2", "marker_block_id": "/page/17/Text/3"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0226", "section": "C.2.2. ROUTING TEST IN SIMILARITY GRAPHS", "page_start": 18, "page_end": 18, "type": "Text", "text": "Due to its simplicity and ease of implementation, CEOs has been adopted in a variety of similarity search tasks (Pham, 2021; Andoni et al., 2015; Xu & Pham, 2024). Beyond standalone similarity estimation, CEOs has also been leveraged to accelerate similarity graphs, which constitute one of the most effective structures for ANNS. By swapping the roles of the query and data vectors in the original CEOs formulation, Lu et al. (2024) developed a space-partitioning technique and proposed the PEOs test. This test enables probabilistic comparisons between the objective angle and a fixed threshold, and has been integrated into the routing procedures of similarity graphs. Specifically, for every visited node u, we send all the out-neighbors of u to PEOs. Only the exact distances between q and the nodes that pass the PEOs test are computed. In the experiments of (Lu et al., 2024), only around 25% nodes can pass the PEOs test. As a result, substantial improvements in search performance brought by PEOs were reported over similarity graphs such as HNSW (Malkov & Yashunin, 2020) and NSSG (Fu et al., 2022). More recently, Lu et al. (2025) proposed a new test function, KS2, which achieves higher test accuracy while maintaining a shorter test time compared with PEOs. KS2 employs a projection structure similar to that of PEOs, but additionally incorporates the reference angle into the testing procedure. Lu et al. (2025) further showed that, without introducing additional assumptions, the test guarantees a success probability of at least 0.5 when deciding whether the exact distance between the objective node and q should be evaluated.", "source": "marker_v2", "marker_block_id": "/page/17/Text/5"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0227", "section": "C.2.2. ROUTING TEST IN SIMILARITY GRAPHS", "page_start": 18, "page_end": 18, "type": "TableGroup", "text": "Table 4. PAG parameter settings. Dataset PAG-Base (efC, M, L) PAG-Lite (efC, M, L) Modern Datasets DBpedia1536 DBpedia3072 (1000, 32, 128) (1000, 32, 64) (100, 32, 128) (100, 32, 64) WoltFood (2000, 128, 64) (100, 32, 04) (100, 64, 32) DataCompDr AmazonBooks (1000, 32, 64) (2000, 64, 64) (100, 32, 64) (100, 64, 32) MajorTOM (1000, 32, 64) (100, 04, 32) (100, 16, 64) Legacy Datasets Word2Vec (8000, 64, 32) (200, 64, 32) GIST (1000, 64, 96) (64, 32, 96) GloVe (2000, 64, 32) (100, 32, 32) ImageNet (2000, 64, 16) (200, 16, 16) SIFT10M (1000, 32, 16) (100, 16, 16) DEEP100M (1000, 32, 16) (100, 16, 16)", "source": "marker_v2", "marker_block_id": "/page/17/TableGroup/389"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0228", "section": "D.1. Environment", "page_start": 18, "page_end": 18, "type": "Text", "text": "All experiments were conducted on a machine equipped with an Intel Xeon Platinum 8276L CPU, which supports AVX-512 instructions and provides 112 hardware threads. The system was configured with 754 GB of DDR4 ECC memory and runs Ubuntu 22.04. For indexing, all 112 threads were used, while search was performed using a single CPU thread, in line with the standard setting in ANN-Benchmarks (Bernhardsson et al., 2015).", "source": "marker_v2", "marker_block_id": "/page/17/Text/10"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0229", "section": "D.2.1. SELECTION OF BASELINES", "page_start": 18, "page_end": 18, "type": "Text", "text": "For quantization-based methods, we choose IVF-PQFS (Jégou et al., 2011), ScaNN (Guo et al., 2020a), and RabitQ+ (Gao et al., 2025) as baselines, where RabitQ+ (Gao et al., 2025) is an improved version of RabitQ (Gao & Long, 2024). For graph-based methods, we choose HNSW (Malkov & Yashunin, 2020), Vamana (Subramanya et al., 2019), and SymphonyQG (Gou et al., 2025), where SymphonyQG has shown its superiority over LVQ (Aguerrebere et al., 2023) and NGT-QG (Yahoo! Japan, 2023). On the other hand, HNSW+KS2 (Lu et al., 2025) has been shown to perform better than HNSW+PEOs (Lu et al., 2024), and KS2 can be approximately viewed as our PRT without TFB, modulo threshold-setting differences. Thus, for a fair comparison, we re-implement the KS2 component within our framework and report results for PAG versus PRT only in our ablation study.", "source": "marker_v2", "marker_block_id": "/page/17/Text/13"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0230", "section": "D.2.2. PARAMETER SETTING", "page_start": 18, "page_end": 18, "type": "Text", "text": "For graph-based methods, we used large graph construction parameters to ensure their best search performance. To ensure a fair comparison of indexing time, the efC parameter in PAG-Base was set to a similar or even larger value than HNSW.", "source": "marker_v2", "marker_block_id": "/page/17/Text/15"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0231", "section": "D.2.2. PARAMETER SETTING", "page_start": 19, "page_end": 19, "type": "FigureGroup", "text": "Figure 12. QPS-recall of PAG-Base under varying space partition size L, K = 100. M and efC values are given in Table 4, PAG-Base.", "source": "marker_v2", "marker_block_id": "/page/18/FigureGroup/262"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0232", "section": "D.2.2. PARAMETER SETTING", "page_start": 19, "page_end": 19, "type": "FigureGroup", "text": "Figure 13. Indexing time and peak memory usage of PAG-Base under varying space partition size L.\\ M and efC values are given in Table 4, PAG-Base.", "source": "marker_v2", "marker_block_id": "/page/18/FigureGroup/263"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0233", "section": "D.2.2. PARAMETER SETTING", "page_start": 19, "page_end": 19, "type": "FigureGroup", "text": "Figure 14. QPS-recall of PAG-Base under varying (M, efC), K = 100. L values are given in Table 4, PAG-Base.", "source": "marker_v2", "marker_block_id": "/page/18/FigureGroup/264"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0234", "section": "D.2.2. PARAMETER SETTING", "page_start": 19, "page_end": 19, "type": "FigureGroup", "text": "Figure 15. Indexing time and peak memory usage of PAG-Base under varying (M, efC). L values are given in Table 4, PAG-Base.", "source": "marker_v2", "marker_block_id": "/page/18/FigureGroup/265"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0235", "section": "D.2.2. PARAMETER SETTING", "page_start": 19, "page_end": 19, "type": "Text", "text": "(1) HNSW : efC=1024. M=32. The only exception is MajorTOM, where efC=512 due to the long indexing time on this dataset.", "source": "marker_v2", "marker_block_id": "/page/18/Text/268"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0236", "section": "D.2.2. PARAMETER SETTING", "page_start": 19, "page_end": 19, "type": "ListGroup", "text": "(2) Vamana : By default, R=64. L=1024. \\alpha=1.2 . For better QPS-recall performance on MajorTOM and DEEP100M, we choose R=100 and L=250 on MajorTOM, and R=110 and L=200 on DEEP100M. (3) SymQG : efC = 1024. There are two options for the degree, 32 and 64, resulting in two baselines, SymQG (32)", "source": "marker_v2", "marker_block_id": "/page/18/ListGroup/266"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0237", "section": "D.2.2. PARAMETER SETTING", "page_start": 19, "page_end": 19, "type": "ListGroup", "text": "and SymQG (64). (4) ScaNN : We adopt the recommended settings from its GitHub repository (Guo et al., 2020b). (5) RabitQ+ : Following the same setting as in (Gao et al., 2025), b=8.\\ k=16,384 for DataCompDr and 4096 for the other datasets. (6) IVFPQFS : We test combinations of nlist \\in \\{1024, 4096, 16384\\}, k_-factor \\in \\{64, 128, 256\\}, and", "source": "marker_v2", "marker_block_id": "/page/18/ListGroup/267"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0238", "section": "D.2.2. PARAMETER SETTING", "page_start": 20, "page_end": 20, "type": "Text", "text": "M candidates ∈ [48, 384]. A combination is chosen for good QPS-recall performance on each dataset.", "source": "marker_v2", "marker_block_id": "/page/19/Text/1"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0239", "section": "D.2.2. PARAMETER SETTING", "page_start": 20, "page_end": 20, "type": "Text", "text": "(7) PAG: PAG has two versions, PAG-Base and PAG-Lite. Their parameter settings are listed in Table 4. Users can adjust the values of efC, M, and L by taking into account the indexing time, memory footprint, and search efficiency.", "source": "marker_v2", "marker_block_id": "/page/19/Text/2"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0240", "section": "D.2.2. PARAMETER SETTING", "page_start": 20, "page_end": 20, "type": "Text", "text": "other three datasets. PAG-Base also uses less memory than SymQG. The QPS-recall performance of PAG-Lite is generally better than HNSW and Vamana. Meanwhile, PAG-Lite achieves the smallest indexing time, and is competitive in memory footprint.", "source": "marker_v2", "marker_block_id": "/page/19/Text/3"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0241", "section": "E.1. Effect of Space Partition Size L", "page_start": 20, "page_end": 20, "type": "Text", "text": "From the QPS-recall results in Fig. 12, we can see that, within a moderate range, increasing L leads to better search performance, at the cost of a higher memory footprint. As shown in Fig. 13 (left), larger L values also results in longer indexing time, due to the faster search speed for each inserted vector. On the other hand, this causes a moderately larger index size, which is reflected in the memory footprint reported in Fig. 13 (middle and right). As analyzed in Appendix B, L = √ d is an appropriate choice.", "source": "marker_v2", "marker_block_id": "/page/19/Text/6"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0242", "section": "E.2. Effect of M and efC", "page_start": 20, "page_end": 20, "type": "Text", "text": "We vary parameters M and efC and plot the QPS-recall in Fig. 14. Larger M and efC value result in faster query processing speed, and the advantage is more remarkable when users require high recall values. As shown in Fig. 15, increasing M and efC generally leads to more indexing time as well as larger memory footprint. This is expected, because they control the out-degree and the number of nodes visited for each insertion. The only exception is DataCompDr, where a largerM does not necessarily mean a slower indexing speed. This is because the search speed for each inserted vector is accelerated despite a larger out-degree.", "source": "marker_v2", "marker_block_id": "/page/19/Text/8"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0243", "section": "E.3. Evaluation on Additional Legacy Datasets", "page_start": 20, "page_end": 20, "type": "Text", "text": "We report the results on four additional legacy datasets, Word2Vec, ImageNet, GIST, and SIFT10M, which are commonly used for ANNS evaluation. Figs. 16, 17, and 18 show the QPS-recall performances when K = 10, 100, and 1000. Fig. 19 shows the indexing time and memory footprint. From the results in figures, we have the following observations. PAG-Base maintains its competitiveness in QPS-recall performance, as we have witnessed in the experiments on other datasets. On Word2Vec and ImageNet, where graph-based methods and SymQG struggle to achieve high recalls, PAG-Base has a significant advantage in QPS. On GIST and SIFT10M, SymQG performs better than PAG-Base in QPS when K = 10 and K = 100, because the two datasets are sparse and well-suited for vector quantization. When K = 1000, PAG-Base outperforms SymQG by a large margin. For indexing speed, PAG-Base is slower than graphbased methods and SymQG on Word2Vec but is faster on the", "source": "marker_v2", "marker_block_id": "/page/19/Text/10"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0244", "section": "E.3. Evaluation on Additional Legacy Datasets", "page_start": 21, "page_end": 21, "type": "FigureGroup", "text": "Figure 16. QPS-recall on additional legacy datasets, K = 10.", "source": "marker_v2", "marker_block_id": "/page/20/FigureGroup/285"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0245", "section": "E.3. Evaluation on Additional Legacy Datasets", "page_start": 21, "page_end": 21, "type": "FigureGroup", "text": "Figure 17. QPS-recall on additional legacy datasets, K = 100.", "source": "marker_v2", "marker_block_id": "/page/20/FigureGroup/286"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0246", "section": "E.3. Evaluation on Additional Legacy Datasets", "page_start": 21, "page_end": 21, "type": "FigureGroup", "text": "Figure 18. QPS-recall on additional legacy datasets, K = 1000.", "source": "marker_v2", "marker_block_id": "/page/20/FigureGroup/287"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0247", "section": "E.3. Evaluation on Additional Legacy Datasets", "page_start": 21, "page_end": 21, "type": "FigureGroup", "text": "Figure 19. Indexing time and peak memory usage on additional legacy datasets.", "source": "marker_v2", "marker_block_id": "/page/20/FigureGroup/288"}

icml26/fddf30e3-e5ae-4a68-b862-daa6e531883a/appendix_text_v3.txt

@@ -0,0 +1,317 @@
+[p. 15 | section: A. Frequently Used Notations | type: TableGroup]
+Table 3 shows the notations frequently used in this paper. Table 3. Frequently used notations. Symbol Description \mathcal{D} Dataset of vectors d Dimension of vectors n Dataset size \boldsymbol{q} Query vector K Retrieval size \boldsymbol{v} Node to be inserted to a graph index N_{\rm out}(\boldsymbol{v}) Out-neighbors of v N_{\rm in}(\boldsymbol{v}) In-neighbors of \boldsymbol{v} \boldsymbol{u} Candidate neighbor of v to be evaluated L Space partition size \{\boldsymbol{w_i}\}_{i=1}^t t neighbors of \boldsymbol{u} \{\boldsymbol{e_i}\}_{i=1}^t Edges between \boldsymbol{u} and \{\boldsymbol{w_i}\}_{i=1}^t \tau_i Threshold w.r.t. w_i in PRT \delta_i Threshold w.r.t. w_i in PES R_L Result list W Working set R_F, R_T Dual rings of false positives and ejected nodes m Number of projection vectors in each subspace {\mathcal F} Set of concatenated projection vectors \{\boldsymbol{r_i}\}_{i=1}^{m^L} Individual projection vectors r_i^* \in \mathcal{F} Vector having the smallest angle with w_i - u \cos \alpha_i Cosine of the angle between w_i - u and v - u \cos \beta_i Cosine of the angle between w_i - u and r_i^* \cos \theta_i Cosine of the angle between v-u and r_i^* Cand(v) All nodes visited during the insertion of v H(\boldsymbol{v}) Nodes in \operatorname{Cand}(\boldsymbol{v}) rejected by PES
+
+[p. 15 | section: B. Proof of Theorem 3.1 | type: Text]
+In the proof, with slight abuse of notation, we treat u as the origin, and directly use v and each w_i to denote v-u and w_i-u , respectively.
+
+[p. 15 | section: B. Proof of Theorem 3.1 | type: Text]
+As such, we have w_1, \ldots, w_t, v \in \mathbb{S}^{d-1} , and \langle w_i, v \rangle = \cos \alpha_i, i = 1, \ldots, t . Suppose that d is divisible by L. Let w_i = [w_{i1}, \ldots, w_{iL}]^{\top} and v = [v_1, \ldots, v_L]^{\top} be the equal-dimension partition of w_i and v, respectively. Suppose that in the l-th subspace, we generate m random projection vectors \{r_{jl}\}_{j=1}^m with the norm 1/\sqrt{L} . We use r_{il}^* to denote the nearest projection vector to w_{il} , i.e.,
+
+[p. 15 | section: B. Proof of Theorem 3.1 | type: Equation]
+r_{il}^* = \underset{1 \le j \le m}{\arg\max} \langle r_{jl}, w_{il} \rangle. \tag{7}
+
+[p. 15 | section: B. Proof of Theorem 3.1 | type: Text]
+Then, we have \bm{r_i^*} = [\bm{r_{i1}^*}, \dots, \bm{r_{iL}^*}]^\top \in \mathbb{S}^{d-1}. We introduce
+
+[p. 15 | section: B. Proof of Theorem 3.1 | type: Equation]
+C_{il} := \frac{\langle \boldsymbol{w_{il}}, \boldsymbol{r_{il}^*} \rangle \sqrt{L}}{\|\boldsymbol{w_{il}}\|} \in [-1, 1]. \tag{8}
+
+[p. 15 | section: B. Proof of Theorem 3.1 | type: Text]
+Let \mathbb{E}[C_{il}] = \mu_m and \mathrm{Var}(C_{il}) = \sigma_m^2 , where \mu_m and \sigma_m^2 depend only on m. Note that they implicitly depend on the subspace dimension d/L. When m grows at a sufficiently high rate compared to d/L, \mu_m \to 1 and \sigma_m \to 0 . Moreover,
+
+[p. 15 | section: B. Proof of Theorem 3.1 | type: Text]
+we have
+
+[p. 15 | section: B. Proof of Theorem 3.1 | type: Equation]
+Cov(C_{il}, C_{jl}) = \rho_m(\phi_{ijl})\sigma_m^2 \tag{9}
+
+[p. 15 | section: B. Proof of Theorem 3.1 | type: Text]
+where \phi_{ijl} denotes the angle between w_{il} and w_{jl} , and \rho_m := [0,\pi] \to [-1,1] denotes the correlation coefficient function depending on \phi_{ijl} . Then, we have
+
+[p. 15 | section: B. Proof of Theorem 3.1 | type: Equation]
+Y_i := \Sigma_{l=1}^L \langle \boldsymbol{r_{il}^*}, \boldsymbol{w_{il}} \rangle = \Sigma_{l=1}^L \frac{\|\boldsymbol{w_{il}}\|}{\sqrt{L}} C_{il}. (10)
+
+[p. 15 | section: B. Proof of Theorem 3.1 | type: Text]
+By the assumption that \|\boldsymbol{w_{il}}\| = (1 + o(1))/\sqrt{L} , we have
+
+[p. 15 | section: B. Proof of Theorem 3.1 | type: Equation]
+\mathbb{E}[Y_i] = \mu_m(1 + o(1)). \tag{11}
+
+[p. 15 | section: B. Proof of Theorem 3.1 | type: Equation]
+Var(Y_i) = \frac{\sigma_m^2}{L}(1 + o(1)). (12)
+
+[p. 15 | section: B. Proof of Theorem 3.1 | type: Equation]
+Cov(Y_i, Y_j) = \frac{\sigma_m^2}{L^2} \sum_{l=1}^{L} \rho_m(\phi_{ijl}) (1 + o(1)). (13)
+
+[p. 15 | section: B. Proof of Theorem 3.1 | type: Text]
+Then, v_l can be decomposed as v_l = v_{il}^{\parallel} + v_{il}^{\perp} , where v_{il}^{\parallel} and v_{il}^{\perp} are defined as follows.
+
+[p. 15 | section: B. Proof of Theorem 3.1 | type: Equation]
+v_{il}^{\parallel} = \frac{\langle v_l, w_{il} \rangle}{\|w_{il}\|^2} w_{il}, \quad v_{il}^{\perp} \perp w_{il}. (14)
+
+[p. 15 | section: B. Proof of Theorem 3.1 | type: Text]
+Then, we can define Z_i as follows.
+
+[p. 15 | section: B. Proof of Theorem 3.1 | type: Equation]
+Z_i := \langle r_i^*, v \rangle = Z_i^{(1)} + Z_i^{(2)} (15)
+
+[p. 15 | section: B. Proof of Theorem 3.1 | type: Text]
+where Z_i^{(1)} and Z_i^{(2)} are defined as follows.
+
+[p. 15 | section: B. Proof of Theorem 3.1 | type: Equation]
+Z_i^{(1)} := \sum_{l=1}^{L} \frac{\langle \boldsymbol{v_l}, \boldsymbol{w_{il}} \rangle}{\|\boldsymbol{w_{il}}\| \sqrt{L}} C_{il}. (16)
+
+[p. 15 | section: B. Proof of Theorem 3.1 | type: Equation]
+Z_i^{(2)} := \sum_{l=1}^L \langle r_{il}^*, v_{il}^{\perp} \rangle. \tag{17}
+
+[p. 15 | section: B. Proof of Theorem 3.1 | type: Equation]
+\mathbb{E}[Z_i^{(1)}] = \frac{\mu_m}{\sqrt{L}} \sum_{l=1}^L \frac{\langle \boldsymbol{v}_l, \boldsymbol{w}_{il} \rangle}{\|\boldsymbol{w}_{il}\|} = \mu_m \cos \alpha_i + o(1). (18)
+
+[p. 15 | section: B. Proof of Theorem 3.1 | type: Text]
+By symmetry, we have \mathbb{E}[Z_i^{(2)}|\{C_{il}\}_{l=1}^L]=0 . Then, by the independence of subspaces, we have
+
+[p. 15 | section: B. Proof of Theorem 3.1 | type: Equation]
+\mathbb{E}[Z_i] = \mu_m \cos(\alpha_i) + o(1). \tag{19}
+
+[p. 15 | section: B. Proof of Theorem 3.1 | type: Equation]
+Var(Z_i^{(1)}) = \frac{\sigma_m^2}{L} \sum_{l=1}^{L} \frac{\langle v_l, w_{il} \rangle^2}{\|w_{il}\|^2}. (20)
+
+[p. 15 | section: B. Proof of Theorem 3.1 | type: Text]
+We take the orthogonal decomposition of r_{il}^* as r_{il}^* = r_{il}^{\parallel} + r_{il}^{\perp} , where r_{il}^{\parallel} has the same direction with w_{il} . Let r_{il}^{\perp} = \|r_{il}^{\perp}\|\zeta , where \|\zeta\| = 1 is a random vector in a (d/L - 1)-dimensional subspace. We have
+
+[p. 15 | section: B. Proof of Theorem 3.1 | type: Equation]
+\mathbb{E}_{\zeta}[\langle \zeta, v_{il}^{\perp} \rangle^{2}] = \frac{\|v_{il}^{\perp}\|^{2} L}{d - L}. (21)
+
+[p. 16 | section: B. Proof of Theorem 3.1 | type: Text]
+By symmetry, \mathbb{E}[\langle r_{il}^{\perp}, v_{il}^{\perp} \rangle \mid C_{il}] = 0 . Then, we have
+
+[p. 16 | section: B. Proof of Theorem 3.1 | type: Equation]
+\operatorname{Var}(\langle \boldsymbol{r}_{il}^*, \boldsymbol{v}_{il}^{\perp} \rangle) = \mathbb{E}_{C_{il}} \left[ \mathbb{E}[\langle \boldsymbol{r}_{il}^{\perp}, \boldsymbol{v}_{il}^{\perp} \rangle^2 \mid C_{il}] \right]. \tag{22}
+
+[p. 16 | section: B. Proof of Theorem 3.1 | type: Text]
+Using \mathbb{E}[\|\boldsymbol{r}_{il}^{\perp}\|^2] = \frac{1}{L}(1 - \mathbb{E}[C_{il}^2]) , we have
+
+[p. 16 | section: B. Proof of Theorem 3.1 | type: Equation]
+Var(Z_i^{(2)}) = \frac{1 - (\sigma_m^2 + \mu_m^2)}{d - L} \sum_{l=1}^{L} \| v_{il}^{\perp} \|^2. (23)
+
+[p. 16 | section: B. Proof of Theorem 3.1 | type: Text]
+Because Cov(Z_i^{(1)}, Z_i^{(2)}) = 0 , we have
+
+[p. 16 | section: B. Proof of Theorem 3.1 | type: Equation]
+Var(Z_i) = Var(Z_i^{(1)}) + Var(Z_i^{(2)}). (24)
+
+[p. 16 | section: B. Proof of Theorem 3.1 | type: Text]
+We define \epsilon_m:=\sqrt{1-\mathbb{E}[C_{il}^2]} . From Eq. (23), we can see that \mathrm{Var}(Z_i^{(2)})=O(\epsilon_m^2/L) .
+
+[p. 16 | section: B. Proof of Theorem 3.1 | type: Text]
+We now analyze the covariance structure. We cannot assume Z^{(1)} and Z^{(2)} are uncorrelated across different indices i \neq j . Instead, we decompose the covariance matrix as
+
+[p. 16 | section: B. Proof of Theorem 3.1 | type: Equation]
+Cov(Z_i, Z_j) = Cov(Z_i^{(1)}, Z_j^{(1)}) + R_{ij} (25)
+
+[p. 16 | section: B. Proof of Theorem 3.1 | type: Text]
+where the residual term R_{ij} contains the noise auto-covariance and cross-terms:
+
+[p. 16 | section: B. Proof of Theorem 3.1 | type: Equation]
+R_{ij} = \text{Cov}(Z_i^{(2)}, Z_j^{(2)}) + \text{Cov}(Z_i^{(1)}, Z_j^{(2)}) + \text{Cov}(Z_i^{(2)}, Z_j^{(1)}). (26)
+
+[p. 16 | section: B. Proof of Theorem 3.1 | type: Text]
+Because {\rm Var}(Z^{(1)})=O(1/L) and {\rm Var}(Z^{(2)})=O(\epsilon_m^2/L) , by the Cauchy-Schwarz inequality, we have
+
+[p. 16 | section: B. Proof of Theorem 3.1 | type: Equation]
+|\text{Cov}(Z_i^{(1)}, Z_j^{(2)})| \le \sqrt{\text{Var}(Z_i^{(1)})\text{Var}(Z_j^{(2)})} = O\left(\frac{\epsilon_m}{L}\right).
+
+[p. 16 | section: B. Proof of Theorem 3.1 | type: Text]
+Thus, the entire residual term satisfies R_{ij} = O(\epsilon_m/L) .
+
+[p. 16 | section: B. Proof of Theorem 3.1 | type: Text]
+Now, let us turn to \mathrm{Cov}(Y_i,Z_j) . When i=j, by conditional expectation, \mathrm{Cov}(Y_i,Z_j^{(2)})=0 . When i\neq j , if m\to\infty , \sigma_m^2+\mu_m^2\to 1 and \mathrm{Var}(Z_i^{(2)})\to 0 . Thus, \mathrm{Cov}(Y_i,Z_j^{(2)})\to 0 . Then, as m\to\infty , we have
+
+[p. 16 | section: B. Proof of Theorem 3.1 | type: Equation]
+Cov(Y_i, Z_j) \to \frac{\sigma_m^2}{L} \sum_{l=1}^{L} \rho_m(\phi_{ijl}) \langle \boldsymbol{v_l}, \boldsymbol{w_{jl}} \rangle (1 + o(1)). (28)
+
+[p. 16 | section: B. Proof of Theorem 3.1 | type: Text]
+Based on the above analysis, we consider
+
+[p. 16 | section: B. Proof of Theorem 3.1 | type: Equation]
+\boldsymbol{\xi_{l}} = \left[ \frac{\|\boldsymbol{w_{1l}}\|}{\sqrt{L}} C_{1l}, \dots, \frac{\|\boldsymbol{w_{tl}}\|}{\sqrt{L}} C_{tl}, \langle \boldsymbol{r_{1l}^*}, \boldsymbol{v_l} \rangle, \dots, \langle \boldsymbol{r_{tl}^*}, \boldsymbol{v_l} \rangle \right]. (29)
+
+[p. 16 | section: B. Proof of Theorem 3.1 | type: Text]
+By the assumptions that \|v_{il}\| and \|w_{il}\| equal (1 + o(1))/\sqrt{L} , it can be seen that \xi_l satisfies the Lyapunov condition. By the independence of difference subspaces, we use the Lindeberg-Feller CLT and obtain the following result.
+
+[p. 16 | section: B. Proof of Theorem 3.1 | type: Equation]
+\sqrt{L}([Y_1, \dots, Y_t, Z_1, \dots, Z_t]^{\top} - \bar{\boldsymbol{\mu}}) \xrightarrow[L \to \infty]{d} \mathcal{N}(0, \bar{\Sigma}_{m,L}) (30)
+
+[p. 16 | section: B. Proof of Theorem 3.1 | type: Text]
+where \bar{\mu} = [\mu_m, \dots, \mu_m, \mu_m \cos \alpha_1, \dots, \mu_m \cos \alpha_t]^\top , and \bar{\Sigma} is represented by block sub-matrices
+
+[p. 16 | section: B. Proof of Theorem 3.1 | type: Equation]
+\bar{\Sigma} = \begin{pmatrix} \Sigma_{YY} & \Sigma_{YZ} \\ \Sigma_{ZY} & \Sigma_{ZZ} \end{pmatrix} \tag{31}
+
+[p. 16 | section: B. Proof of Theorem 3.1 | type: Text]
+where the elements of \Sigma_{YY} , \Sigma_{YZ} and \Sigma_{ZZ} are L times the covariance computed in Eqs. (13), (28), and (25), respectively.
+
+[p. 16 | section: B. Proof of Theorem 3.1 | type: Text]
+Based on this result, Z | Y is still a multivariate normal distribution. We have
+
+[p. 16 | section: B. Proof of Theorem 3.1 | type: Equation]
+\mathbb{E}[\mathbf{Z}|\mathbf{Y}] = \boldsymbol{\mu}_Z + \boldsymbol{\Sigma}_{ZY} \boldsymbol{\Sigma}_{YY}^{-1} (\mathbf{Y} - \boldsymbol{\mu}_Y). \tag{32}
+
+[p. 16 | section: B. Proof of Theorem 3.1 | type: Equation]
+Cov(\mathbf{Z}|\mathbf{Y}) = (\Sigma_{ZZ} - \Sigma_{ZY}\Sigma_{YY}^{-1}\Sigma_{YZ})/L. (33)
+
+[p. 16 | section: B. Proof of Theorem 3.1 | type: Text]
+Let D be the diagonal matrix defined as follows
+
+[p. 16 | section: B. Proof of Theorem 3.1 | type: Equation]
+\mathbf{D} = \operatorname{diag}[\cos \alpha_1, \dots, \cos \alpha_t]. \tag{34}
+
+[p. 16 | section: B. Proof of Theorem 3.1 | type: Text]
+Because \Sigma_{ZY} \to \mathbf{D}\Sigma_{YY} , when \mathbf{Y} = [\cos \beta_1, \dots, \cos \beta_t]^\top , we have
+
+[p. 16 | section: B. Proof of Theorem 3.1 | type: Equation]
+\mathbb{E}[\mathbf{Z}|\mathbf{Y}] \to [\cos \alpha_1 \cos \beta_1, \dots, \cos \alpha_t \cos \beta_t]^\top. (35)
+
+[p. 16 | section: B. Proof of Theorem 3.1 | type: Equation]
+Cov(\mathbf{Z}|\mathbf{Y}) \to (\mathbf{\Sigma}_{\mathbf{Z}\mathbf{Z}} - \mathbf{D}\mathbf{\Sigma}_{\mathbf{Y}\mathbf{Y}}\mathbf{D})/L = \mathbf{R}/L (36)
+
+[p. 16 | section: B. Proof of Theorem 3.1 | type: Text]
+where \mathbf{R} represents the scaled residual matrix with entries L \cdot R_{ij} = O(\epsilon_m) . Then, we conclude.
+
+[p. 16 | section: B. Proof of Theorem 3.1 | type: Text]
+Remarks. For each fixed i, if we apply a random rotation matrix to w_i - u and v - u in \mathbb{R}^d , by the spherical concentration inequality, we can derive the following inequalities.
+
+[p. 16 | section: B. Proof of Theorem 3.1 | type: Equation]
+P\left(\left|\|(\boldsymbol{w_i} - \boldsymbol{u})_l\|^2 - \frac{1}{L}\right| \ge \tilde{\epsilon}\right) \le 2e^{-c_1 d\tilde{\epsilon}^2} (37)
+
+[p. 16 | section: B. Proof of Theorem 3.1 | type: Equation]
+P\left(\left|\langle (\boldsymbol{v}-\boldsymbol{u})_l, (\boldsymbol{w_i}-\boldsymbol{u})_l \rangle - \frac{\cos \alpha_i}{L}\right| \ge \tilde{\epsilon}\right) \le 2e^{-c_2 d\tilde{\epsilon}^2} (38)
+
+[p. 16 | section: B. Proof of Theorem 3.1 | type: Text]
+where c_1 and c_2 are constants. As L \to \infty and d/(L^2) \to \infty , we have the following results:
+
+[p. 16 | section: B. Proof of Theorem 3.1 | type: Equation]
+\|(\boldsymbol{w_i} - \boldsymbol{u})_l\| = (1 + o_p(1))/\sqrt{L}. (39)
+
+[p. 16 | section: B. Proof of Theorem 3.1 | type: Equation]
+\langle (\boldsymbol{v} - \boldsymbol{u})_l, (\boldsymbol{w_i} - \boldsymbol{u})_l \rangle = \frac{\cos \alpha}{L} + o_p(\frac{1}{L}). (40)
+
+[p. 16 | section: B. Proof of Theorem 3.1 | type: Text]
+This implies that L=\sqrt{d} is a balanced choice, which is consistent with the setting in our experiments.
+
+[p. 16 | section: C. Related Work | type: Text]
+We first supplement Sec. 1.1 with more discussions, and then introduce the preliminaries on probabilistic routing.
+
+[p. 17 | section: C.1. Discussions on ANNS Solvers | type: Text]
+911912
+
+[p. 17 | section: C.1.1. VECTOR QUANTIZATION | type: Text]
+Quantization-based methods for ANNS have a long history. Representative approaches can be found in (Jégou et al., 2011; Ge et al., 2014; Babenko & Lempitsky, 2016; Guo et al., 2020a; Gao & Long, 2024; Gao et al., 2025). Here, we briefly introduce the basic idea of the most widely-used method, Product Quantization (PQ) (Jégou et al., 2011).
+
+[p. 17 | section: C.1.1. VECTOR QUANTIZATION | type: Text]
+Let \mathcal{D} \subset \mathbb{R}^d be a d-dimensional dataset and let q \in \mathbb{R}^d be a query vector. PQ divides each data vector x \in \mathcal{D} into L sub-vectors, i.e., x = (x_1, x_2, \ldots, x_L) , where each sub-vector x_j \in \mathbb{R}^{d'} has dimension d' = d/L. In this way, PQ constructs L subspaces of dimension d', each containing the corresponding sub-vectors of all data points.
+
+[p. 17 | section: C.1.1. VECTOR QUANTIZATION | type: Text]
+In each subspace, every sub-vector is assigned to one of 2^k quantized sub-vectors. Equivalently, each sub-vector is represented by a sub-codeword of length k bits. The set of all sub-codewords in the j-th subspace is referred to as a codebook and is denoted by \mathcal{C}_j . By concatenating the sub-codewords across all subspaces, each original vector is represented by a codeword of length kL, taking values in the Cartesian product \mathcal{C}_1 \times \cdots \times \mathcal{C}_L .
+
+[p. 17 | section: C.1.1. VECTOR QUANTIZATION | type: Text]
+During the query phase, PQ first computes the distances between the query \boldsymbol{q} and all quantized sub-vectors in each subspace. For each data vector \boldsymbol{x} , PQ then sums the corresponding L subspace distances to obtain an approximate distance between \boldsymbol{x} and \boldsymbol{q} . After computing the approximate distances for all data vectors, PQ ranks them accordingly and returns the most promising candidates.
+
+[p. 17 | section: C.1.2. SIMILARITY GRAPH | type: Text]
+The workflow of graph-based methods is roughly as follows. In the indexing phase, a similarity graph is constructed as the index structure, where data points serve as nodes and edges connect pairs of nearby nodes. A search can move directly from one node to another only if an edge exists between them. In the query phase, the node with the highest priority in a priority queue is selected, and all of its connected neighbors are visited. During this process, the priority queue is updated whenever a node closer to the query is discovered. The search ends when all nodes in the priority queue have been visited. Finally, the top-K points in the priority queue are returned.
+
+[p. 17 | section: C.1.2. SIMILARITY GRAPH | type: Text]
+Among various graph-based methods, HNSW (Malkov & Yashunin, 2020) is a notable one that employs a multi-layer hierarchical structure to achieve rapid routing. NSG (Fu et al., 2019) optimizes the graph topology to ensure the existence of monotonic paths towards a central entry point, thereby enhancing search efficiency. Vamana, which was introduced along with DiskANN (Subramanya et al., 2019), iteratively refines a random graph into a high-performance graph struc-
+
+[p. 17 | section: C.1.2. SIMILARITY GRAPH | type: Text]
+ture. Despite their structural differences, these methods converge on a similar edge selection criterion, namely RobustPrune , which prioritizes directional diversity over simple proximity to ensure efficient navigation through high-dimensional space.
+
+[p. 17 | section: C.1.2. SIMILARITY GRAPH | type: Text]
+In addition to the three graphs mentioned above, many graph-based methods have been proposed recently (Lu et al., 2021; Gao & Long, 2023; Xie et al., 2025; Wang et al., 2025b). Despite different designs, most of them rely on existing graphs, such as HNSW.
+
+[p. 17 | section: C.1.3. QUANTIZED GRAPH | type: Text]
+As stated in the main body of this paper, QG can achieve high performance on some datasets for small values of K, and in certain cases can be 4 \times -10 \times faster than HNSW. However, such improvement is often not robust and suffers from several limitations: (1) Sensitivity to data distribution. The effectiveness of QG relies on the assumption that the ranking induced by quantized distances is sufficiently close to the ranking under the original distances. Unfortunately, this assumption often does not hold, especially for many modern real-world datasets where semantic embeddings are increasingly complex. (2) Sensitivity to K. As K increases, the performance of QG degrades significantly. When K reaches the thousand scale, QG generally has no significant advantage over HNSW. (3) Very large space cost. To improve the accuracy of QG, more bits are required to represent the quantized vectors. As a result, the memory consumption of QG is typically at least 2 \times larger than HNSW, compromising the use of QG for large datasets.
+
+[p. 17 | section: C.2.1. RANDOM PROJECTION FOR ANGLE ESTIMATION | type: Text]
+Because the norms can be pre-computed, estimating the \ell_2 distance is equivalent to estimating the cosine of the angle between vectors. In high-dimensional Euclidean spaces, angle estimation via random-projection techniques has been extensively studied, with Locality Sensitive Hashing (LSH) (Indyk & Motwani, 1998; Andoni & Indyk, 2005; 2008) being one of the most influential approaches. Among various LSH methods, SimHash (Charikar, 2002) is a representative one. Its core idea is to generate multiple random hyperplanes that partition the space into cells, so that vectors falling into the same cell are likely to form a small angle with each other. Subsequent studies have proposed more refined strategies for angular distance estimation. In particular, Andoni et al. (2015) introduced Falconn, an LSH method that identifies the projection vector yielding the largest or smallest projection value for a given data vector and uses the corresponding projection index as the hash value. This design leads to substantially improved search performance compared to SimHash. Building on this idea, Pham (2021) further incorporated Con-
+
+[p. 18 | section: C.2.1. RANDOM PROJECTION FOR ANGLE ESTIMATION | type: Text]
+938939
+
+[p. 18 | section: C.2.1. RANDOM PROJECTION FOR ANGLE ESTIMATION | type: Text]
+944945946
+
+[p. 18 | section: C.2.1. RANDOM PROJECTION FOR ANGLE ESTIMATION | type: Text]
+949950
+
+[p. 18 | section: C.2.1. RANDOM PROJECTION FOR ANGLE ESTIMATION | type: Caption]
+Figure 11. DataCompDr data (sampled 50,000 vectors) and query (1,000 vectors) distributions.
+
+[p. 18 | section: C.2.1. RANDOM PROJECTION FOR ANGLE ESTIMATION | type: Text]
+comitants of Extreme Order Statistics (CEOs) to explicitly identify the projection achieving the maximum or minimum inner product with the data vector, and to record the associated extreme projected value. By exploiting this additional information beyond a discrete hash value, a more accurate estimation of angular distance can be achieved (Pham & Liu, 2022).
+
+[p. 18 | section: C.2.2. ROUTING TEST IN SIMILARITY GRAPHS | type: Text]
+Due to its simplicity and ease of implementation, CEOs has been adopted in a variety of similarity search tasks (Pham, 2021; Andoni et al., 2015; Xu & Pham, 2024). Beyond standalone similarity estimation, CEOs has also been leveraged to accelerate similarity graphs, which constitute one of the most effective structures for ANNS. By swapping the roles of the query and data vectors in the original CEOs formulation, Lu et al. (2024) developed a space-partitioning technique and proposed the PEOs test. This test enables probabilistic comparisons between the objective angle and a fixed threshold, and has been integrated into the routing procedures of similarity graphs. Specifically, for every visited node u, we send all the out-neighbors of u to PEOs. Only the exact distances between q and the nodes that pass the PEOs test are computed. In the experiments of (Lu et al., 2024), only around 25% nodes can pass the PEOs test. As a result, substantial improvements in search performance brought by PEOs were reported over similarity graphs such as HNSW (Malkov & Yashunin, 2020) and NSSG (Fu et al., 2022). More recently, Lu et al. (2025) proposed a new test function, KS2, which achieves higher test accuracy while maintaining a shorter test time compared with PEOs. KS2 employs a projection structure similar to that of PEOs, but additionally incorporates the reference angle into the testing procedure. Lu et al. (2025) further showed that, without introducing additional assumptions, the test guarantees a success probability of at least 0.5 when deciding whether the exact distance between the objective node and q should be evaluated.
+
+[p. 18 | section: C.2.2. ROUTING TEST IN SIMILARITY GRAPHS | type: TableGroup]
+Table 4. PAG parameter settings. Dataset PAG-Base (efC, M, L) PAG-Lite (efC, M, L) Modern Datasets DBpedia1536 DBpedia3072 (1000, 32, 128) (1000, 32, 64) (100, 32, 128) (100, 32, 64) WoltFood (2000, 128, 64) (100, 32, 04) (100, 64, 32) DataCompDr AmazonBooks (1000, 32, 64) (2000, 64, 64) (100, 32, 64) (100, 64, 32) MajorTOM (1000, 32, 64) (100, 04, 32) (100, 16, 64) Legacy Datasets Word2Vec (8000, 64, 32) (200, 64, 32) GIST (1000, 64, 96) (64, 32, 96) GloVe (2000, 64, 32) (100, 32, 32) ImageNet (2000, 64, 16) (200, 16, 16) SIFT10M (1000, 32, 16) (100, 16, 16) DEEP100M (1000, 32, 16) (100, 16, 16)
+
+[p. 18 | section: D.1. Environment | type: Text]
+All experiments were conducted on a machine equipped with an Intel Xeon Platinum 8276L CPU, which supports AVX-512 instructions and provides 112 hardware threads. The system was configured with 754 GB of DDR4 ECC memory and runs Ubuntu 22.04. For indexing, all 112 threads were used, while search was performed using a single CPU thread, in line with the standard setting in ANN-Benchmarks (Bernhardsson et al., 2015).
+
+[p. 18 | section: D.2.1. SELECTION OF BASELINES | type: Text]
+For quantization-based methods, we choose IVF-PQFS (Jégou et al., 2011), ScaNN (Guo et al., 2020a), and RabitQ+ (Gao et al., 2025) as baselines, where RabitQ+ (Gao et al., 2025) is an improved version of RabitQ (Gao & Long, 2024). For graph-based methods, we choose HNSW (Malkov & Yashunin, 2020), Vamana (Subramanya et al., 2019), and SymphonyQG (Gou et al., 2025), where SymphonyQG has shown its superiority over LVQ (Aguerrebere et al., 2023) and NGT-QG (Yahoo! Japan, 2023). On the other hand, HNSW+KS2 (Lu et al., 2025) has been shown to perform better than HNSW+PEOs (Lu et al., 2024), and KS2 can be approximately viewed as our PRT without TFB, modulo threshold-setting differences. Thus, for a fair comparison, we re-implement the KS2 component within our framework and report results for PAG versus PRT only in our ablation study.
+
+[p. 18 | section: D.2.2. PARAMETER SETTING | type: Text]
+For graph-based methods, we used large graph construction parameters to ensure their best search performance. To ensure a fair comparison of indexing time, the efC parameter in PAG-Base was set to a similar or even larger value than HNSW.
+
+[p. 19 | section: D.2.2. PARAMETER SETTING | type: FigureGroup]
+Figure 12. QPS-recall of PAG-Base under varying space partition size L, K = 100. M and efC values are given in Table 4, PAG-Base.
+
+[p. 19 | section: D.2.2. PARAMETER SETTING | type: FigureGroup]
+Figure 13. Indexing time and peak memory usage of PAG-Base under varying space partition size L.\ M and efC values are given in Table 4, PAG-Base.
+
+[p. 19 | section: D.2.2. PARAMETER SETTING | type: FigureGroup]
+Figure 14. QPS-recall of PAG-Base under varying (M, efC), K = 100. L values are given in Table 4, PAG-Base.
+
+[p. 19 | section: D.2.2. PARAMETER SETTING | type: FigureGroup]
+Figure 15. Indexing time and peak memory usage of PAG-Base under varying (M, efC). L values are given in Table 4, PAG-Base.
+
+[p. 19 | section: D.2.2. PARAMETER SETTING | type: Text]
+(1) HNSW : efC=1024. M=32. The only exception is MajorTOM, where efC=512 due to the long indexing time on this dataset.
+
+[p. 19 | section: D.2.2. PARAMETER SETTING | type: ListGroup]
+(2) Vamana : By default, R=64. L=1024. \alpha=1.2 . For better QPS-recall performance on MajorTOM and DEEP100M, we choose R=100 and L=250 on MajorTOM, and R=110 and L=200 on DEEP100M. (3) SymQG : efC = 1024. There are two options for the degree, 32 and 64, resulting in two baselines, SymQG (32)
+
+[p. 19 | section: D.2.2. PARAMETER SETTING | type: ListGroup]
+and SymQG (64). (4) ScaNN : We adopt the recommended settings from its GitHub repository (Guo et al., 2020b). (5) RabitQ+ : Following the same setting as in (Gao et al., 2025), b=8.\ k=16,384 for DataCompDr and 4096 for the other datasets. (6) IVFPQFS : We test combinations of nlist \in \{1024, 4096, 16384\}, k_-factor \in \{64, 128, 256\}, and
+
+[p. 20 | section: D.2.2. PARAMETER SETTING | type: Text]
+M candidates ∈ [48, 384]. A combination is chosen for good QPS-recall performance on each dataset.
+
+[p. 20 | section: D.2.2. PARAMETER SETTING | type: Text]
+(7) PAG: PAG has two versions, PAG-Base and PAG-Lite. Their parameter settings are listed in Table 4. Users can adjust the values of efC, M, and L by taking into account the indexing time, memory footprint, and search efficiency.
+
+[p. 20 | section: D.2.2. PARAMETER SETTING | type: Text]
+other three datasets. PAG-Base also uses less memory than SymQG. The QPS-recall performance of PAG-Lite is generally better than HNSW and Vamana. Meanwhile, PAG-Lite achieves the smallest indexing time, and is competitive in memory footprint.
+
+[p. 20 | section: E.1. Effect of Space Partition Size L | type: Text]
+From the QPS-recall results in Fig. 12, we can see that, within a moderate range, increasing L leads to better search performance, at the cost of a higher memory footprint. As shown in Fig. 13 (left), larger L values also results in longer indexing time, due to the faster search speed for each inserted vector. On the other hand, this causes a moderately larger index size, which is reflected in the memory footprint reported in Fig. 13 (middle and right). As analyzed in Appendix B, L = √ d is an appropriate choice.
+
+[p. 20 | section: E.2. Effect of M and efC | type: Text]
+We vary parameters M and efC and plot the QPS-recall in Fig. 14. Larger M and efC value result in faster query processing speed, and the advantage is more remarkable when users require high recall values. As shown in Fig. 15, increasing M and efC generally leads to more indexing time as well as larger memory footprint. This is expected, because they control the out-degree and the number of nodes visited for each insertion. The only exception is DataCompDr, where a largerM does not necessarily mean a slower indexing speed. This is because the search speed for each inserted vector is accelerated despite a larger out-degree.
+
+[p. 20 | section: E.3. Evaluation on Additional Legacy Datasets | type: Text]
+We report the results on four additional legacy datasets, Word2Vec, ImageNet, GIST, and SIFT10M, which are commonly used for ANNS evaluation. Figs. 16, 17, and 18 show the QPS-recall performances when K = 10, 100, and 1000. Fig. 19 shows the indexing time and memory footprint. From the results in figures, we have the following observations. PAG-Base maintains its competitiveness in QPS-recall performance, as we have witnessed in the experiments on other datasets. On Word2Vec and ImageNet, where graph-based methods and SymQG struggle to achieve high recalls, PAG-Base has a significant advantage in QPS. On GIST and SIFT10M, SymQG performs better than PAG-Base in QPS when K = 10 and K = 100, because the two datasets are sparse and well-suited for vector quantization. When K = 1000, PAG-Base outperforms SymQG by a large margin. For indexing speed, PAG-Base is slower than graphbased methods and SymQG on Word2Vec but is faster on the
+
+[p. 21 | section: E.3. Evaluation on Additional Legacy Datasets | type: FigureGroup]
+Figure 16. QPS-recall on additional legacy datasets, K = 10.
+
+[p. 21 | section: E.3. Evaluation on Additional Legacy Datasets | type: FigureGroup]
+Figure 17. QPS-recall on additional legacy datasets, K = 100.
+
+[p. 21 | section: E.3. Evaluation on Additional Legacy Datasets | type: FigureGroup]
+Figure 18. QPS-recall on additional legacy datasets, K = 1000.
+
+[p. 21 | section: E.3. Evaluation on Additional Legacy Datasets | type: FigureGroup]
+Figure 19. Indexing time and peak memory usage on additional legacy datasets.

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+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0003", "section": "1. Introduction", "page_start": 1, "page_end": 1, "type": "Text", "text": "Seeing the fast development of AI technology, we argue that a well-designed ANNS solver should perform well on modern datasets, and beyond (D1) QPS-recall performance, several demands must also be considered: (D2) indexing time, for which graph indexes such as HNSW (Malkov & Yashunin, 2020) are often criticized (Lin, 2025; Li & Papakonstantinou, 2025) , compromising their use in applications that require instant deployment; (D3) memory footprint, where moderate and adjustable consumption, mostly caused by the index size, is desired for the memory vs accuracy trade-off (Cheng et al., 2024; Wallace, 2025) ; (D4) scalability to high dimensionality, as motivated by the increasing dimensionality of modern embedding models such as CLIP (Radford et al., 2021) 1 ; (D5) robustness against the retrieval size K, due to the needs in various applications (e.g., K is typically 10 for RAG (Fensore et al., 2025; Ke et al., 2025) but can be up to hundreds in image retrieval (Vendrow et al., 2024) and thousands in recommender systems (Zhang et al., 2025) ); (D6) support of online insertions, which is essential for emergent applications such as self-evolving agents that continually accumulate and reuse experience through interaction (Ouyang et al., 2025; Zhai et al., 2025; Zhang et al., 2026) .", "source": "marker_v2", "marker_block_id": "/page/0/Text/10"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0004", "section": "1.1. Prior Works", "page_start": 1, "page_end": 1, "type": "Text", "text": "We review representative ANNS solvers based on D1 – D6. More discussions on literature can be found in Appendix C.", "source": "marker_v2", "marker_block_id": "/page/0/Text/12"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0005", "section": "1.1. Prior Works", "page_start": 1, "page_end": 1, "type": "Footnote", "text": "1 Despite the availability of dimensionality reduction techniques, Weller et al. (2025) proved that for embedding-based retrieval with dimensionality d, there exists a dataset size n (e.g., n ≈ 4M for d = 1024) beyond which it is theoretically impossible to represent all possible top-2 document combinations, showcasing the necessity for higher dimensionality on large datasets.", "source": "marker_v2", "marker_block_id": "/page/0/Footnote/13"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0006", "section": "1.1. Prior Works", "page_start": 2, "page_end": 2, "type": "TableGroup", "text": "Table 1. Summary of empirical performance comparison of notable ANNS solvers, HNSW (Malkov & Yashunin, 2020), Vamana (in-memory DiskANN (Subramanya et al., 2019)), IVFPQFS (fast scan IVFPQ (Jégou et al., 2011)), ScaNN (Guo et al., 2020a), RaBitQ+ (Gao et al., 2025), SymQG (Gou et al., 2025), HNSW+KS2 (Lu et al., 2025), and our solutions PAG-Base (for high QPS) and PAG-Lite (for fast indexing and small index size). Detailed results are available in Sec. 5.2. The evaluation of D1 – D3 is clustered into tiers, where Tier-4 is the best. D4 – D6 are evaluated in yes/no (√/X). Graph-Based Quantization-Based QG PG PA ıG Criteria HNSW Vamana IVFPQFS 1 ScaNN 1 RaBitQ+ SymQG HNSW+KS2 PAG-Base PAG-Lite D1. QPS-recall Tier-2 Tier-2 Tier-1 Tier-1 Tier-1 Tier-3 Tier-3 Tier-4 Tier-3 D2. Indexing time Tier-1 Tier-1 Tier-3 Tier-3 Tier-3 Tier-2 Tier-1 Tier-2 Tier-4 D3. Memory footprint 2 Tier-3 Tier-3 Tier-4 Tier-4 Tier-4 Tier-1 Tier-2 Tier-3 Tier-4 D4. High-dim. scalability ✓ ✓ ✓ ✓ ✓ X ✓ ✓ ✓ D5. Retrieval size robustness ✓ ✓ ✓ ✓ ✓ Х ✓ ✓ ✓ D6. Online insertion support 3 ✓ ✓ ✓ ✓ \\checkmark X × ✓ ✓ Reranking is enabled for IVFPOFS and ScaNN.", "source": "marker_v2", "marker_block_id": "/page/1/TableGroup/346"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0007", "section": "1.1. Prior Works", "page_start": 2, "page_end": 2, "type": "Text", "text": "Graph-Based Methods. These methods construct a similarity graph connecting nearby vectors and hop towards the neighborhood of q. Notable methods are HNSW (Malkov & Yashunin, 2020), NSG (Fu et al., 2019), and DiskANN (Subramanya et al., 2019). They are generally competitive in QPS-recall but slow in building indexes.", "source": "marker_v2", "marker_block_id": "/page/1/Text/7"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0008", "section": "1.1. Prior Works", "page_start": 2, "page_end": 2, "type": "Text", "text": "Quantization-Based Methods. These methods compress vectors and rank them using approximate (i.e., quantized) distances to q in pursuit of efficiency. Representative methods are IVFPQ (Jégou et al., 2011) and ScaNN (Guo et al., 2020a). They are fast in building indexes and save memory, but their QPS-recall is generally inferior to graph-based methods.", "source": "marker_v2", "marker_block_id": "/page/1/Text/8"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0009", "section": "1.1. Prior Works", "page_start": 2, "page_end": 2, "type": "Text", "text": "Projection-Based Methods. These methods use random (e.g., E2LSH (Andoni & Indyk, 2005), Falconn (Andoni et al., 2015), and CEOs (Pham, 2021)) or data-dependent (e.g., learning to hash (Wang et al., 2018)) projections of the original vectors for indexing. Although many of them enjoy theoretical guarantees, they are less competitive than graphand quantization-based methods in QPS-recall performance, hence becoming less popular in modern top-K ANNS.", "source": "marker_v2", "marker_block_id": "/page/1/Text/9"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0010", "section": "1.1. Prior Works", "page_start": 2, "page_end": 2, "type": "Text", "text": "<u>Tree-Based Methods.</u> They utilize tree indexes. table methods include k-d tree (Bentley, 1990), cover tree (Beygelzimer et al., 2006), and Annoy (Bernhardsson, 2013). Like projection-based methods, they are less widely used for modern ANNS due to limited QPS-recall performance.", "source": "marker_v2", "marker_block_id": "/page/1/Text/10"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0011", "section": "1.1. Prior Works", "page_start": 2, "page_end": 2, "type": "Text", "text": "As discussed above, graph-based methods are good at QPSrecall. But they compute exact distances between vectors, which is computationally costly. To address this weakness, recent advancements attempt to integrate other techniques, in particular, quantization or projection, into graphs.", "source": "marker_v2", "marker_block_id": "/page/1/Text/11"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0012", "section": "1.1. Prior Works", "page_start": 2, "page_end": 2, "type": "Text", "text": "Quantized Graph (QG) Methods. They construct a similarity graph based on quantized vectors instead of the original ones, thereby replacing exact distances with approximate values. Representative methods are NGT-QG (Yahoo! Japan, 2023),", "source": "marker_v2", "marker_block_id": "/page/1/Text/12"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0013", "section": "1.1. Prior Works", "page_start": 2, "page_end": 2, "type": "Text", "text": "LVQ (Aguerrebere et al., 2023), and SymphonyQG (Gou et al., 2025). QG methods can achieve very high QPS-recall performance, yet they are sensitive to the data distribution and many of them do not perform very well in D3 - D6.", "source": "marker_v2", "marker_block_id": "/page/1/Text/13"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0014", "section": "1.1. Prior Works", "page_start": 2, "page_end": 2, "type": "Text", "text": "Projection + Graph ( PG ) Methods . Unlike QG, they operate on original vectors and reduce unnecessary exact distance computations. To this end, they employ projection techniques to test whether a neighbor needs to be explored (called routing tests). Representative methods are FINGER (Chen et al., 2023), PEOs (Lu et al., 2024), and KS2 (Lu et al., 2025). Despite QPS-recall improvement on top of graph-based methods, such improvement comes at the cost of indexing time, memory footprint, and support for online insertions.", "source": "marker_v2", "marker_block_id": "/page/1/Text/14"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0015", "section": "1.1. Prior Works", "page_start": 2, "page_end": 2, "type": "Text", "text": "Table 1 compares notable methods implemented in leading vector databases and recent ones representing state of the art.", "source": "marker_v2", "marker_block_id": "/page/1/Text/15"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0016", "section": "1.2. Our Solution", "page_start": 2, "page_end": 2, "type": "Text", "text": "We propose a new ANNS framework, Projection-Augmented Graph (PAG), which integrates projection into a similarity graph and achieves superior performance for all the six criteria. Unlike PG methods, PAG treats projection as a fundamental building block of graph construction rather than as a plug-in. The rationale of PAG is to accommodate both exact and approximate distances within a unified framework, addressing two key issues: (1) when distances shall be computed exactly and when they shall be approximated, and (2) how exact and approximate distances are compared during indexing and searching to enhance the performance.", "source": "marker_v2", "marker_block_id": "/page/1/Text/17"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0017", "section": "1.2. Our Solution", "page_start": 2, "page_end": 2, "type": "Text", "text": "PAG significantly reduces the computational cost of searching (D1) and indexing (D2) by carefully determining whether exact distance computation is needed. Such procedure relies on asymmetric comparisons between exact and approximate distance values obtained from space-efficient and adjustable random-projection structures (D3), as opposed to symmetric comparisons in QG. Moreover, the asymmetric distance", "source": "marker_v2", "marker_block_id": "/page/1/Text/18"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0018", "section": "1.2. Our Solution", "page_start": 2, "page_end": 2, "type": "Footnote", "text": "Memory footprint for searching, rather than indexing, is compared.", "source": "marker_v2", "marker_block_id": "/page/1/Footnote/5"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0019", "section": "1.2. Our Solution", "page_start": 2, "page_end": 2, "type": "Footnote", "text": "& lt;sup>3</sup> For any streaming workload Q composed of search and insertion queries, we say an algorithm supports online insertion, if the amortized cost of processing a query in Q is O(1) times the cost of a search. In other words, the index can be incrementally updated with minimal cost. See Sec. 4.3 for the analysis of PAG.", "source": "marker_v2", "marker_block_id": "/page/1/Footnote/6"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0020", "section": "1.2. Our Solution", "page_start": 3, "page_end": 3, "type": "Text", "text": "comparisons, as guided by random-projection-based statistical tests, only need to tell which value is larger, thereby achieving accuracy while preserving efficiency. Meanwhile, PAG inherits graph-based methods' scalability to high dimensionality (D4) and robustness against retrieval size (D5), and follows the search-and-insertion paradigm (e.g., HNSW) to accommodate online insertions (D6).", "source": "marker_v2", "marker_block_id": "/page/2/Text/1"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0021", "section": "1.2. Our Solution", "page_start": 3, "page_end": 3, "type": "Text", "text": "Our technical contributions are summarized as follows.", "source": "marker_v2", "marker_block_id": "/page/2/Text/2"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0022", "section": "1.2. Our Solution", "page_start": 3, "page_end": 3, "type": "ListGroup", "text": "(1) Following the idea of routing tests in PEOs (Lu et al., 2024) and KS2 (Lu et al., 2025) , we derive a Probabilistic Routing Test (PRT) function (Sec. 3.2) . The theoretical result (Theorem 3.1) , along with Lemma 4.3 in Lu et al. (2025) , provides a complete theoretical explanation of PRT. In addition, our work is the first to apply PRT to graph construction. (2) We propose a data structure called Test Feedback Buffer (TFB) (Sec. 3.3) , which refines the threshold setting in PRT and enables the reuse of false positives generated by PRT. By incorporating TFB into PRT, we obtain the PRT-TFB test as a core technique for accelerating both indexing and searching. (3) We propose a statistical test called Probabilistic Edge Selection (PES) (Sec. 3.4) , which is derived from Theorem 3.1 and can expand in-degrees when necessary. In collaboration with PRT, PRT-PES can improve the search performance for hard datasets on which traditional graph indexes perform poorly, while incurring very small indexing overhead. (4) PRT, TFB, and PES constitute PAG (Fig. 1) . We show how the three components interact within PAG, with implementation details and complexity analysis provided. (5) We conduct experiments on six modern (post-2023) datasets covering text, image, and multimodal data, with dimensionality ranging from 384 to 3072 and retrieval size from 10 to 1000. The results show that PAG achieves the best QPS-recall performance (up to 5× faster than HNSW), and its superiority is particularly evident on datasets with higher dimensionality. PAG remains dominant as K increases. By adjusting parameters, PAG can deliver the fastest indexing speed and lowest memory footprint on most datasets while maintaining competitive QPS-recall. PAG also performs well on the datasets in the ANN-Benchmarks, showcasing its compatibility with data generated by legacy models.", "source": "marker_v2", "marker_block_id": "/page/2/ListGroup/782"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0023", "section": "2. Problem Setting", "page_start": 3, "page_end": 3, "type": "Text", "text": "Since PAG adopts a search-and-insertion paradigm, its search process, similar to HNSW, can be seen as a special case of index construction (i.e., no update to the graph), and we can focus on the problems that arise in index construction. Because constructing a similarity graph is essentially determining its edge set, we consider the following question: Can we design statistical methods that select edges efficiently (i.e., fast indexing) and effectively (i.e., fast searching)? To an-", "source": "marker_v2", "marker_block_id": "/page/2/Text/9"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0024", "section": "2. Problem Setting", "page_start": 3, "page_end": 3, "type": "Text", "text": "swer this question, we first review the existing edge-selection strategy, and then formalize the problems to be solved.", "source": "marker_v2", "marker_block_id": "/page/2/Text/10"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0025", "section": "2.1. Background of RobustPrune", "page_start": 3, "page_end": 3, "type": "Text", "text": "Given a directed graph, let v be a node whose edges are to be determined. To identify its in-neighbor set Nin(v) and out-neighbor set Nout(v), state-of-the-art methods, such as HNSW, NSG and Vamana, employ a strategy known as RobustPrune to decide whether a candidate edge should be preserved. Take the construction of Nin(v) as an example. For a candidate neighbor u of node v, the pruning criterion considers the following set 2 :", "source": "marker_v2", "marker_block_id": "/page/2/Text/12"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0026", "section": "2.1. Background of RobustPrune", "page_start": 3, "page_end": 3, "type": "Equation", "text": "S_{u,v} = \\{ w \\in N_{\\text{out}}(u) \\mid ||w - v|| \\le ||v - u|| \\}. (1)", "source": "marker_v2", "marker_block_id": "/page/2/Equation/13"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0027", "section": "2.1. Background of RobustPrune", "page_start": 3, "page_end": 3, "type": "Text", "text": "In RobustPrune, if there exists w in Su, v such that ∥w − u∥ ≤ ∥v − u∥, u is not added to Nin(v); otherwise, Nin(v) is updated as Nin(v)∪{u}. This criterion admits an intuitive interpretation: if ∥w − u∥ ≤ ∥v − u∥ holds for some w, it is likely that one can reach v from u via w along a monotonic search path (Fu et al., 2019) , implying that the direct edge uv⃗ is redundant. Similarly, Nout(v) can be constructed by RobustPrune with the roles of u and v reversed. In sum, existing graph-based methods determine Nout(v) and Nin(v) sequentially in the following manner.", "source": "marker_v2", "marker_block_id": "/page/2/Text/14"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0028", "section": "2.1. Background of RobustPrune", "page_start": 3, "page_end": 3, "type": "Text", "text": "Nout(v) . We take v as the query and conduct an ANNS from v to obtain a candidate set, which is stored in a priority queue P. RobustPrune is then applied to P to determine Nout(v).", "source": "marker_v2", "marker_block_id": "/page/2/Text/15"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0029", "section": "2.1. Background of RobustPrune", "page_start": 3, "page_end": 3, "type": "Text", "text": "Nin(v) . We check every u in Nout(v) and preserve those u's retained by RobustPrune as Nin(v).", "source": "marker_v2", "marker_block_id": "/page/2/Text/16"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0030", "section": "2.2. Towards Fast Graph Construction", "page_start": 3, "page_end": 3, "type": "Text", "text": "During graph construction, most of the time is spent on ANNS for the candidate set of Nout(v). Although we can reduce the graph construction parameter (e.g., efC in HNSW) for fast graph construction, the query processing performance degrades accordingly. Thus, we raise the following question.", "source": "marker_v2", "marker_block_id": "/page/2/Text/18"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0031", "section": "2.2. Towards Fast Graph Construction", "page_start": 3, "page_end": 3, "type": "Text", "text": "Q1. How can we accelerate the ANNS for v while preserving the query processing performance?", "source": "marker_v2", "marker_block_id": "/page/2/Text/19"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0032", "section": "2.2. Towards Fast Graph Construction", "page_start": 3, "page_end": 3, "type": "Text", "text": "Existing studies PEOs and KS2 have shown that PRT has the potential to be an appropriate answer to Q1. For each visited node u and its out-neighbor w, PRT sends (u, v, w) to a probabilistic test function to see whether they can pass the test, and then determines whether the exact distance between v and w needs to be computed based on the test result. Consequently, PRT avoids unnecessary distance computations, leading to faster search. In KS2 (Lu et al., 2025) , it is shown that the routing test under ℓ 2 and cosine distances is equivalent to the angle-thresholding problem when vector norms", "source": "marker_v2", "marker_block_id": "/page/2/Text/20"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0033", "section": "2.2. Towards Fast Graph Construction", "page_start": 3, "page_end": 3, "type": "Footnote", "text": "For Vamana, we assume its pruning ratio α = 1 and omit it in the inequality.", "source": "marker_v2", "marker_block_id": "/page/2/Footnote/21"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0034", "section": "2.2. Towards Fast Graph Construction", "page_start": 4, "page_end": 4, "type": "FigureGroup", "text": "Figure 1. An overview of PAG. In this example, \\boldsymbol{u} has 7 out-neighbors \\{\\boldsymbol{w_i}\\}_{i=1}^7 . Let \\{\\boldsymbol{e_i}\\}_{i=1}^7 denote the edges between \\boldsymbol{u} and \\{\\boldsymbol{w_i}\\}_{i=1}^7 , which are sent to PRT, where the threshold is determined by \\boldsymbol{z_{max}} . As a result, only \\boldsymbol{w_3} and \\boldsymbol{w_5} pass PRT, and their exact distances to \\boldsymbol{v} are computed. By distance comparison, node \\boldsymbol{w_5} is identified as a false positive, not added to W, and thus sent to R_F . Node \\boldsymbol{w_3} is inserted into W, causing \\boldsymbol{z_{max}} to be ejected from W and sent to R_T . The two rings R_F and R_T are merged and refilled into W. On the other route, by a left shift, the signs of both \\boldsymbol{w_3} and \\boldsymbol{w_5} are reversed, and all signs become negative, indicating that \\boldsymbol{u} is rejected by the PES test. Consequently, \\boldsymbol{u}\\dot{\\boldsymbol{v}} is treated as a candidate edge and added to the PES set.", "source": "marker_v2", "marker_block_id": "/page/3/FigureGroup/339"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0035", "section": "2.2. Towards Fast Graph Construction", "page_start": 4, "page_end": 4, "type": "Text", "text": "are stored explicitly. Therefore, PRT can be formulated as follows:", "source": "marker_v2", "marker_block_id": "/page/3/Text/4"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0036", "section": "2.2. Towards Fast Graph Construction", "page_start": 4, "page_end": 4, "type": "Text", "text": "Problem 2.1. ( Probabilistic Routing Test ) Given u, v, w \\in \\mathbb{S}^{d-1} and a threshold -1 \\le \\tau \\le 1 , construct a random-projection vector set \\mathcal{F} and design a test function w.r.t \\mathcal{F} , i.e., PRT : (u, v, w, \\tau) \\to RV with time complexity o(d), where RV denotes the set of one-dimensional random variables, such that if \\cos(\\angle(v-u, w-u)) \\ge \\tau , then \\mathbb{P}[\\operatorname{PRT}(u, v, w, \\tau) \\ge 0] \\ge 0.5 . Otherwise, \\mathbb{P}[\\operatorname{PRT}(u, v, w, \\tau) < 0] \\to 1 as |\\mathcal{F}| \\to \\infty .", "source": "marker_v2", "marker_block_id": "/page/3/Text/5"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0037", "section": "2.2. Towards Fast Graph Construction", "page_start": 4, "page_end": 4, "type": "Text", "text": "Different from the routing test functions used in PEOs and KS2, we take \\tau as an explicit input. In this paper, we use another method to determine \\tau in order to further improve the efficiency of the routing test.", "source": "marker_v2", "marker_block_id": "/page/3/Text/6"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0038", "section": "2.3. Towards High Graph Connectivity", "page_start": 4, "page_end": 4, "type": "Text", "text": "From the process of determining N_{\\rm in}(\\boldsymbol{v}) , we observe that its candidate set is restricted to N_{\\rm out}(\\boldsymbol{v}) . Recent work (Wang et al., 2025a) shows that |N_{\\rm out}(\\boldsymbol{v})| may be too small to reliably determine incoming edges, causing some nodes to have very small in-degrees. Such nodes may become unreachable during search, which partly explains why HNSW performs poorly on certain real-world datasets. On the other hand, for each checked node \\boldsymbol{u} , the worst-case time complexity of RobustPrune can reach O(|N_{\\rm out}(\\boldsymbol{u})|d) , implying that a straightforward enlargement of the candidate set for", "source": "marker_v2", "marker_block_id": "/page/3/Text/8"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0039", "section": "2.3. Towards High Graph Connectivity", "page_start": 4, "page_end": 4, "type": "FigureGroup", "text": "Figure 2. Illustration of PES (left: the role of PES; right: geometric illustration). Let \\boldsymbol{v} be the node to be inserted. The four green nodes are out-neighbors of \\boldsymbol{v} . Without PES, we can obtain only a single in-neighbor via RobustPrune. By taking all other visited nodes (the blue ones) into account, we apply PES, followed by RobustPrune. As such, we can identify three additional promising in-neighbors, thereby strengthening the connectivity of the neighborhood of \\boldsymbol{v} .", "source": "marker_v2", "marker_block_id": "/page/3/FigureGroup/340"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0040", "section": "2.3. Towards High Graph Connectivity", "page_start": 4, "page_end": 4, "type": "Text", "text": "determining incoming edges is expensive. Considering this dilemma, we raise the following question.", "source": "marker_v2", "marker_block_id": "/page/3/Text/11"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0041", "section": "2.3. Towards High Graph Connectivity", "page_start": 4, "page_end": 4, "type": "Text", "text": "Q2. How can we efficiently detect incoming edges outside N_{\\text{out}}(v) that are useful for improving query processing performance but hard to be identified by RobustPrune?", "source": "marker_v2", "marker_block_id": "/page/3/Text/12"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0042", "section": "2.3. Towards High Graph Connectivity", "page_start": 5, "page_end": 5, "type": "Text", "text": "To answer this question, we formulate a problem as follows.", "source": "marker_v2", "marker_block_id": "/page/4/Text/1"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0043", "section": "2.3. Towards High Graph Connectivity", "page_start": 5, "page_end": 5, "type": "Text", "text": "Problem 2.2. ( Probabilistic Edge Selection ) Given u, v \\in \\mathbb{S}^{d-1} , construct a random-projection-vector set \\mathcal{F} and design a probabilistic edge selection function PES: (u, v) \\to RV whose time complexity is O(|N_{\\mathrm{out}}(u)|) , such that if S_{u,v} is non-empty, then \\mathbb{P}[\\mathrm{PES}(u,v) \\geq 0] \\geq 0.5 . Otherwise, \\mathbb{P}[\\mathrm{PES}(u,v) < 0] \\to 1 as |\\mathcal{F}| \\to \\infty .", "source": "marker_v2", "marker_block_id": "/page/4/Text/2"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0044", "section": "2.3. Towards High Graph Connectivity", "page_start": 5, "page_end": 5, "type": "Text", "text": "We emphasize that the PES test is far more than a fast probabilistic implementation of RobustPrune. This is because the PES test can be applied to all visited nodes during the ANNS of \\boldsymbol{v} , whose number is typically much larger than N_{\\rm out}(\\boldsymbol{v}) in practice, leading to a better graph index in terms of connectivity (Fig. 2).", "source": "marker_v2", "marker_block_id": "/page/4/Text/3"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0045", "section": "3. Projection-Augmented Graph", "page_start": 5, "page_end": 5, "type": "Text", "text": "We show that Problems 2.1 and 2.2 can be solved in a unified framework PAG, which contains three components: Probabilistic Routing Test (PRT), Test Feedback Buffer (TFB), and Probabilistic Edge Selection (PES).", "source": "marker_v2", "marker_block_id": "/page/4/Text/5"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0046", "section": "3.1. Neighborhood Relations via Random Projection", "page_start": 5, "page_end": 5, "type": "Text", "text": "We present an asymptotic result that characterizes the relationship between multiple angles in high-dimensional spaces and their corresponding projection values onto a certain projection vector. This result forms the theoretical basis for all the three components. We first describe how to construct the random-projection vector set \\mathcal F that appears in Problems 2.1 and 2.2, following an approach similar to that used in KS2 (Lu et al., 2025). Specifically, we divide the original space \\mathbb R^d into L subspaces, each of dimension d/L. In each subspace, we generate multiple cross-polytopes and apply an independent rotation to each, producing a total of m normalized vectors on \\mathbb S^{d/L-1} . By concatenation, we obtain a total of m^L normalized vectors \\{r_j\\}_{j=1}^{m^L} , which together form the set \\mathcal F . Consider v as the node to be inserted and v as the candidate neighbor to be checked, let N_{\\mathrm{out}}(u) := \\{w_i\\}_{i=1}^t . We define \\{\\alpha_i\\}_{i=1}^t as follows.", "source": "marker_v2", "marker_block_id": "/page/4/Text/7"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0047", "section": "3.1. Neighborhood Relations via Random Projection", "page_start": 5, "page_end": 5, "type": "Equation", "text": "\\alpha_i := \\arccos \\frac{\\langle \\boldsymbol{w_i} - \\boldsymbol{u}, \\boldsymbol{v} - \\boldsymbol{u} \\rangle}{\\|\\boldsymbol{w_i} - \\boldsymbol{u}\\| \\|\\boldsymbol{v} - \\boldsymbol{u}\\|}, \\quad 1 \\le i \\le t \\quad (2)", "source": "marker_v2", "marker_block_id": "/page/4/Equation/8"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0048", "section": "3.1. Neighborhood Relations via Random Projection", "page_start": 5, "page_end": 5, "type": "Text", "text": "Since \\|\\boldsymbol{w_i} - \\boldsymbol{u}\\| and \\|\\boldsymbol{v} - \\boldsymbol{u}\\| can be pre-computed, we aim to estimate [\\cos \\alpha_1, \\dots, \\cos \\alpha_t]^{\\top} without exact inner product computation in Eq. (2). We will show that this can be realized by \\mathcal{F} . For each \\boldsymbol{w_i} ( 1 \\leq i \\leq t ), let \\boldsymbol{r_i^*} \\in \\mathcal{F} be the reference vector that has the smallest angle with \\boldsymbol{w_i} - \\boldsymbol{u} , and denote this smallest angle by \\beta_i . We use \\cos \\theta_i to denote the cosine of the angle between \\boldsymbol{r_i^*} and \\boldsymbol{v} - \\boldsymbol{u} , and subscript l to denote the l-th sub-vector of the original vector in \\mathbb{R}^d , 1 \\leq l \\leq L . We introduce the following assumptions for each i:", "source": "marker_v2", "marker_block_id": "/page/4/Text/9"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0049", "section": "3.1. Neighborhood Relations via Random Projection", "page_start": 5, "page_end": 5, "type": "Equation", "text": "(A1) \\|(\\boldsymbol{w_i} - \\boldsymbol{u})\\| = 1 and \\|(\\boldsymbol{v} - \\boldsymbol{u})\\| = 1 .", "source": "marker_v2", "marker_block_id": "/page/4/Equation/10"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0050", "section": "3.1. Neighborhood Relations via Random Projection", "page_start": 5, "page_end": 5, "type": "Equation", "text": "(A2) \\|(\\boldsymbol{w_i} - \\boldsymbol{u})_l\\| and \\|(\\boldsymbol{v} - \\boldsymbol{u})_l\\| are equal to (1 + o(1))/\\sqrt{L} . (A3) \\langle (\\boldsymbol{v} - \\boldsymbol{u})_l, (\\boldsymbol{w_i} - \\boldsymbol{u})_l \\rangle = (\\cos \\alpha_i)(1 + o(1))/L .", "source": "marker_v2", "marker_block_id": "/page/4/Equation/11"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0051", "section": "3.1. Neighborhood Relations via Random Projection", "page_start": 5, "page_end": 5, "type": "Text", "text": "Here, A1 does not lose generality, and A2 and A3 are mild for large d and L (see the remarks in Appendix B). Under these assumptions, the following theorem shows that \\{\\alpha_i\\}_{i=1}^t can be asymptotically estimated by \\{\\beta_i\\}_{i=1}^t and \\{\\theta_i\\}_{i=1}^t .", "source": "marker_v2", "marker_block_id": "/page/4/Text/12"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0052", "section": "3.1. Neighborhood Relations via Random Projection", "page_start": 5, "page_end": 5, "type": "Text", "text": "Theorem 3.1. Under A1–A3, as L \\to \\infty , d/L \\to \\infty , and m grows sufficiently fast with respect to d/L, X = [\\cos \\theta_1, \\dots, \\cos \\theta_t]^\\top , conditioned on \\{\\alpha_i, \\beta_i\\}_{i=1}^t , is asymptotically Gaussian:", "source": "marker_v2", "marker_block_id": "/page/4/Text/13"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0053", "section": "3.1. Neighborhood Relations via Random Projection", "page_start": 5, "page_end": 5, "type": "Equation", "text": "X \\mid \\{\\alpha_i, \\beta_i\\}_{i=1}^t \\xrightarrow{d} \\mathcal{N}(\\bar{\\mu}, \\bar{\\Sigma}_{m,L}) (3)", "source": "marker_v2", "marker_block_id": "/page/4/Equation/14"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0054", "section": "3.1. Neighborhood Relations via Random Projection", "page_start": 5, "page_end": 5, "type": "Text", "text": "where the mean is \\bar{\\mu} = [\\cos \\alpha_1 \\cos \\beta_1, \\dots, \\cos \\alpha_t \\cos \\beta_t]^\\top , and \\bar{\\Sigma}_{m,L} = O(\\epsilon_m/L) with \\epsilon_m \\to 0 as m \\to \\infty .", "source": "marker_v2", "marker_block_id": "/page/4/Text/15"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0055", "section": "3.1. Neighborhood Relations via Random Projection", "page_start": 5, "page_end": 5, "type": "Text", "text": "Remarks. (1) Theorem 3.1 establishes a geometric relationship among multiple out-neighbors of \\boldsymbol{u} and \\boldsymbol{v} , allowing us to estimate the relative location of \\boldsymbol{v} w.r.t. \\boldsymbol{u} within the neighborhood of \\boldsymbol{u} . This relationship is the key to both PRT and PES. (2) In our proof, m \\to \\infty is used to eliminate noise from off-diagonal entries, while L \\to \\infty is required for the central limit theorem (CLT). In practice, moderate values of m and L are sufficient. Weak results under finite (m, L) can be found in Lemma 4.3 in the KS2 paper (Lu et al., 2025). (3) \\boldsymbol{X} can be computed efficiently by AVX512 on modern CPUs. Vector [\\cos \\beta_1, \\ldots, \\cos \\beta_t]^{\\top} can be pre-computed, meaning that we can obtain an efficient way to estimate all \\cos \\alpha_i 's simultaneously.", "source": "marker_v2", "marker_block_id": "/page/4/Text/16"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0056", "section": "3.2. Probabilistic Routing Test", "page_start": 5, "page_end": 5, "type": "Text", "text": "Let \\tau_i be a threshold w.r.t w_i . Our PRT function is as follows:", "source": "marker_v2", "marker_block_id": "/page/4/Text/18"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0057", "section": "3.2. Probabilistic Routing Test", "page_start": 5, "page_end": 5, "type": "Equation", "text": "PRT(\\boldsymbol{u}, \\boldsymbol{v}, \\boldsymbol{w_i}, \\tau_i) = \\frac{\\cos \\theta_i}{\\cos \\beta_i} - \\tau_i. (4)", "source": "marker_v2", "marker_block_id": "/page/4/Equation/19"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0058", "section": "3.2. Probabilistic Routing Test", "page_start": 5, "page_end": 5, "type": "Text", "text": "For fixed (u, v), if the value of PRT is positive, the corresponding w_i passes the PRT. Based on Theorem 3.1, under asymptotic assumptions, the PRT function satisfies all the required properties in Problem 2.1. Here, we do not specify the setting of \\tau_i , which will be postponed to Sec. 3.3.", "source": "marker_v2", "marker_block_id": "/page/4/Text/20"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0059", "section": "3.2. Probabilistic Routing Test", "page_start": 5, "page_end": 5, "type": "Text", "text": "Remarks. Following the use of routing test in PEOs and KS2, exact distances are computed only between v and those w_i 's that pass PRT (see Fig. 1). The PRT function has the same structure as the KS2 test function (Lu et al., 2025), except for the setting of the threshold \\tau . However, since the PRT and KS2 tests are derived using different principles, their theoretical results are complementary. Theorem 4.3 in the KS2 paper shows that, for every single w_i , even for finite (m, L), if \\cos(\\angle(v - u, w_i - u)) \\ge \\tau_i , then \\mathbb{P}[\\operatorname{PRT}(u, v, w_i, \\tau_i) \\ge 0] \\ge 0.5 still holds. Our result, on the other hand, characterizes the concrete asymptotic distribution and reveals the impact of L on the covariance, thereby explaining how L is used to adjust the estimation accuracy.", "source": "marker_v2", "marker_block_id": "/page/4/Text/21"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0060", "section": "3.3. Test Feedback Buffer", "page_start": 6, "page_end": 6, "type": "Text", "text": "For the PRT function, a key issue is how to set an appropriate threshold τ . In PEOs and KS2, τ is simply determined by the current furthest point in the result list (priority queue) P. However, due to the existence of false positives (FPs) generated by PRT, some points may pass PRT but cannot be added to P, which implies that the exact distance computations w.r.t such points are redundant. On the other hand, the Gaussian distribution established in Theorem 3.1 implies that, with high probability, the actual distances of these FPs to the query q are not much larger than the threshold τ . This observation naturally raises the following question: can we incrementally increase the threshold τ so that the currently generated FPs can be reused in subsequent search process?", "source": "marker_v2", "marker_block_id": "/page/5/Text/2"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0061", "section": "3.3. Test Feedback Buffer", "page_start": 6, "page_end": 6, "type": "Text", "text": "We give an affirmative answer to this question and propose TFB, which consists of four components: the top-K result list RL, the working set W, and two ring buffers R F and R T , where |W| = |R F | = |R T |. The search procedure is divided into multiple rounds. In the j-th round, we operate on W and adopt the standard Best-First Search (BFS) strategy to update its nodes. During the BFS over W, in addition to the nodes stored in W, there are two types of nodes whose exact distances to the query q are computed. The first type consists of nodes ejected from W; these nodes are inserted into R T . The second type consists of false-positive (FP) nodes that pass the PRT but cannot be added to W; these nodes are inserted into R F . After all nodes in W have been visited, we update R L and clear W. We then merge R F and R T and sort the combined list by distance to q. The sorted nodes are first inserted into W until it is full, and the remaining nodes are inserted into R T . Finally, R F is cleared. The ANNS is completed after several such rounds. In the j-th round, let zmax denote the furthest node in W. τ i in Eq. (4) is set to be the threshold on the cosine of the angle between w i − u and v − u required for w i to be admitted into W, i.e.,", "source": "marker_v2", "marker_block_id": "/page/5/Text/3"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0062", "section": "3.3. Test Feedback Buffer", "page_start": 6, "page_end": 6, "type": "Equation", "text": "\\tau_i = \\frac{\\|u - w_i\\|^2 + \\|u - v\\|^2 - \\|z_{\\max} - v\\|^2}{2\\|v - u\\|\\|w_i - u\\|}. (5)", "source": "marker_v2", "marker_block_id": "/page/5/Equation/4"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0063", "section": "3.3. Test Feedback Buffer", "page_start": 6, "page_end": 6, "type": "Text", "text": "PRT-TFB Test. We call the combination of Eq. (4) and Eq. (5) the PRT-TFB test. Compared with existing routing test methods such as PEOs and KS2, which operate on the whole priority queue P, the PRT-TFB test has the following two advantages. (1) For large efC or large efS—which denotes the result list size, same as the notation in HNSW—the size of W is much smaller than that of P, and the movement of elements in W is much faster than that in P. (2) Each FP has a good chance of being selected as the visited node in the future rounds only if it remains in either of two rings, ensuring that its exact distance to q can be utilized at a certain time point.", "source": "marker_v2", "marker_block_id": "/page/5/Text/5"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0064", "section": "3.4. Probabilistic Edge Selection", "page_start": 6, "page_end": 6, "type": "Text", "text": "In RobustPrune, ∥w i − v∥ needs to be smaller than ∥v − u∥ to allow u to be retained. Let δ i := ∥w i − u∥/(2∥v − u∥) denote the threshold on the cosine of the angle between w i − u and v − u, such that ∥w i − v∥ < ∥v − u∥. Then, the PES function is designed as follows.", "source": "marker_v2", "marker_block_id": "/page/5/Text/7"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0065", "section": "3.4. Probabilistic Edge Selection", "page_start": 6, "page_end": 6, "type": "Equation", "text": "PES(\\boldsymbol{u}, \\boldsymbol{v}) = \\max_{1 \\le i \\le t} \\left( \\frac{\\cos \\theta_i}{\\cos \\beta_i} - \\delta_i \\right). (6)", "source": "marker_v2", "marker_block_id": "/page/5/Equation/8"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0066", "section": "3.4. Probabilistic Edge Selection", "page_start": 6, "page_end": 6, "type": "Text", "text": "By Theorem 3.1, we can see that the PES function in Eq. (6) satisfies the probabilistic properties in Problem 2.2. In practice, if the value in Eq. (6) is negative, we say that (u, v) is rejected by PES, and this edge will be regarded as a promising candidate for further examination of RobustPrune.", "source": "marker_v2", "marker_block_id": "/page/5/Text/9"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0067", "section": "3.4. Probabilistic Edge Selection", "page_start": 6, "page_end": 6, "type": "Text", "text": "PRT-PES Collaboration. From their definitions, the difference between the two thresholds, that is, τ i − δ i , is a value independent of v − w and can be pre-computed. Thus, we can conduct the PES test based on the test results of the PRT, leading to an O(1) time complexity, which confirms the objective O(|Nout(u)|) stated in Problem 2.2.", "source": "marker_v2", "marker_block_id": "/page/5/Text/10"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0068", "section": "3.4. Probabilistic Edge Selection", "page_start": 6, "page_end": 6, "type": "Text", "text": "PES Set. For every u whose PES value is negative, we do not check uv⃗ by RobustPrune immediately. Instead, we add uv⃗ into a so-called PES set for later examination (see Fig. 1) .", "source": "marker_v2", "marker_block_id": "/page/5/Text/11"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0069", "section": "4.1. Implementation", "page_start": 6, "page_end": 6, "type": "Text", "text": "Indexing Phase. PAG inserts every v ∈ D sequentially into the graph. For each v, the process can be divided into the following three steps.", "source": "marker_v2", "marker_block_id": "/page/5/Text/14"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0070", "section": "4.1. Implementation", "page_start": 6, "page_end": 6, "type": "Text", "text": "Step 1. PAG performs ANNS equipped with the PRT-TFB test to obtain the elements stored in RL, and then uses RobustPrune to determine Nout(v) and Nin(v).", "source": "marker_v2", "marker_block_id": "/page/5/Text/15"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0071", "section": "4.1. Implementation", "page_start": 6, "page_end": 6, "type": "Text", "text": "Step 2. Let Cand(v) denote the set of all nodes visited during ANNS. PAG uses PRT-PES to select additional in-neighbors from Cand(v), thereby supplementing the PES set.", "source": "marker_v2", "marker_block_id": "/page/5/Text/16"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0072", "section": "4.1. Implementation", "page_start": 6, "page_end": 6, "type": "Text", "text": "Based on our preceding discussion, PRT-TFB ensures the efficiency of ANNS, while PRT-PES detects more promising edges. We finally select more edges from the PES set.", "source": "marker_v2", "marker_block_id": "/page/5/Text/17"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0073", "section": "4.1. Implementation", "page_start": 6, "page_end": 6, "type": "Text", "text": "Step 3. For each candidate edge in the PES set, we apply RobustPrune to determine if it can be added to the graph.", "source": "marker_v2", "marker_block_id": "/page/5/Text/18"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0074", "section": "4.1. Implementation", "page_start": 6, "page_end": 6, "type": "Text", "text": "Searching Phase. The query processing with PAG is exactly the PRT-TFB-based ANNS for a query q.", "source": "marker_v2", "marker_block_id": "/page/5/Text/19"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0075", "section": "4.1. Implementation", "page_start": 6, "page_end": 6, "type": "Text", "text": "Online Insertion. Like HNSW, PAG naturally supports online insertion. The PES set is checked only after enough new nodes have been inserted into the graph.", "source": "marker_v2", "marker_block_id": "/page/5/Text/20"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0076", "section": "4.2. Algorithm Pseudo-Codes of PAG", "page_start": 7, "page_end": 7, "type": "Text", "text": "The pseudo-codes of the PAG algorithms are presented in Algs. 1 and 2, which represent the indexing and searching, respectively. Note that indexing is essentially the sequential insertion of all nodes. Similar to Lu et al. (2024) , we store the computed projections in a projection table for lookup (Line 8, Alg. 1 and Line 4, Alg. 2) .", "source": "marker_v2", "marker_block_id": "/page/6/Text/3"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0077", "section": "4.2. Algorithm Pseudo-Codes of PAG", "page_start": 7, "page_end": 7, "type": "Text", "text": "24 Add the projection information of new edges to I; 25 Apply RobustPrune to the PES set, supplement Nin(v)'s, and add the projection information of new edges to I;", "source": "marker_v2", "marker_block_id": "/page/6/Text/19"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0078", "section": "4.3. Complexity Analysis", "page_start": 7, "page_end": 7, "type": "Text", "text": "Time complexity. Searching with HNSW is in O(n ′d)-time, where n ′ is the number of visited nodes during the search. Although a rigorous analysis of n ′ remains an open problem, n ′ can be roughly estimated as O(M log n), where 2M is the", "source": "marker_v2", "marker_block_id": "/page/6/Text/5"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0079", "section": "Algorithm 2: Search with PAG", "page_start": 7, "page_end": 7, "type": "Code", "text": "Input: G(D, E, I) is the constructed similarity graph; F is the random-projection structure; efS is the result list size; q is the query; K is the retrieval size Output: Top-K ANN nodes 1 The capacity of RL is set to K; 2 b = max{10, K}; 3 Maximum round Rmax is set to efS/b; 4 Compute all ⟨ql, rjl⟩'s and build a projection table; 5 Insert b elements to W and compute their distance to q; 6 for r = 1 to Rmax do 7 foreach unvisited u ∈ W do 8 foreach w in Nout(u) do 9 if w passes PRT-TFB test then 10 Compute the distance between q and w; 11 if ∥w − q∥ < ∥zmax − q∥ then 12 Update W and insert zmax to RT ; 13 else 14 Insert w to RF ; 15 Update the nodes in W to RL, and empty W; 16 Sort and merge RF and RT , and empty RF and RT ; 17 Refill W until it is full using the sorted elements, and refill RT with the remaining sorted nodes; 18 Return the top-K nodes in RL;", "source": "marker_v2", "marker_block_id": "/page/6/Code/7"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0080", "section": "Algorithm 2: Search with PAG", "page_start": 7, "page_end": 7, "type": "Text", "text": "maximum out-degree. In contrast, the complexity of PAG is O(n ′L+γn′d), whereL ≪ d is the user-specified parameter and γ ≤ 0.5 denotes the ratio of nodes that passed the PRT-PES test relative to the total number of candidates checked. γ is small in practice (generally < 0.2), and is smaller than the ratio in PEOs and KS2, because TFB ensures that the threshold increases incrementally rather than being set to the furthest distance w.r.t. a full priority queue.", "source": "marker_v2", "marker_block_id": "/page/6/Text/8"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0081", "section": "Algorithm 2: Search with PAG", "page_start": 7, "page_end": 7, "type": "Text", "text": "As for indexing, due to the search-and-insertion paradigm, we focus on the insertion of v, for which the complexity is O(n ′L + γn′d + M2d + M dm). Here, O(M2d) is the complexity of node connection, and O(M dm) is used to compute projection information, both of which are much smaller than the ANNS part due to their independence of n ′ .", "source": "marker_v2", "marker_block_id": "/page/6/Text/9"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0082", "section": "Algorithm 2: Search with PAG", "page_start": 7, "page_end": 7, "type": "Text", "text": "Space complexity. The space complexity of searching with PAG is O(nd + nM + nML), where O(nd) corresponds to the original dataset, O(nM) corresponds to the edge set and O(nML) corresponds to the inference structure. In the indexing phase, since we adopt in-place search for insertion, we only require an additional space of at most O(Mn) to store the edges in PES set. To further reduce the memory footprint, we can follow LVQ (Aguerrebere et al., 2023) and represent floats with 2 bytes.", "source": "marker_v2", "marker_block_id": "/page/6/Text/10"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0083", "section": "4.4. Parameter Analysis of PAG", "page_start": 7, "page_end": 7, "type": "Text", "text": "We analyze the settings of the parameters in PAG.", "source": "marker_v2", "marker_block_id": "/page/6/Text/12"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0084", "section": "4.4. Parameter Analysis of PAG", "page_start": 7, "page_end": 7, "type": "Text", "text": "(1) m: The value of m is fixed to 16, so that each projection vector in a subspace is represented by 4 bits, which is com-", "source": "marker_v2", "marker_block_id": "/page/6/Text/13"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0085", "section": "4.4. Parameter Analysis of PAG", "page_start": 8, "page_end": 8, "type": "Text", "text": "Table 2. Dataset statistics.", "source": "marker_v2", "marker_block_id": "/page/7/Text/2"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0086", "section": "4.4. Parameter Analysis of PAG", "page_start": 8, "page_end": 8, "type": "Text", "text": "Name/Source Dataset Size Query Size Dim. OOD Type Embedding Model Distance Measure Modern Datasets DBpedia1536 999,000 1,000 1,536 No Text text-embedding-3-large (OpenAI, 2024) Euclidean DBpedia3072 999,000 1,000 3,072 No Text text-embedding-3-large (OpenAI, 2024) Euclidean WoltFood 1,719,611 1,000 512 No Image clip-ViT-B-32 (Radford et al., 2021) Euclidean DataCompDr 12,779,520 1,000 1,536 Yes Text-to-Image coca ViT-L-14 (Yu et al., 2022) Euclidean AmazonBooks 15,928,208 1,000 384 No Text all-MiniLM-L12-v2 (Wang et al., 2020) Euclidean MajorTOM 56,506,400 10,000 1,024 No Image DINOv2 (Oquab et al., 2024) Euclidean Legacy Datasets Word2Vec 1,000,000 1,000 300 No Text Word2Vec (Mikolov et al., 2013) Euclidean GIST 1,000,000 1,000 960 No Image GIST (Oliva & Torralba, 2001) Euclidean GloVe 1,193,514 1,000 200 No Text GloVe (Pennington et al., 2014) Euclidean ImageNet 2,340,373 200 150 No Image dense SIFT (Lazebnik et al., 2006) Euclidean SIFT10M 10,000,000 1,000 128 No Image SIFT (Lowe, 2004) Cosine DEEP100M 100,000,000 1,000 96 No Image GoogLeNet (Szegedy et al., 2015) Cosine", "source": "marker_v2", "marker_block_id": "/page/7/Text/4"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0087", "section": "4.4. Parameter Analysis of PAG", "page_start": 8, "page_end": 8, "type": "Text", "text": "patible with the AVX512 instruction set. The choice of 4 bits follows NGT-QG (Yahoo! Japan, 2023) .", "source": "marker_v2", "marker_block_id": "/page/7/Text/5"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0088", "section": "4.4. Parameter Analysis of PAG", "page_start": 8, "page_end": 8, "type": "ListGroup", "text": "(2) L: L is recommended to be in [8, d/8], where 8 is compatible with AVX512 because at most 8 levels can be accessed at a time, while d/L = 8 is a safe ratio that ensures the accuracy of the multi-level projection structure, as explained in PEOs (Lu et al., 2024) . Based on the analysis in Sec. 4.3, L is used to tune the trade-off between efficiency and accuracy. (3) M: Similar to its usage in HNSW, 2M is the maximum out-degree of PAG. Based on the analysis in Sec. 4.3, M is recommended to be 64 when space cost permits. In practice, M is generally chosen from {16, 32, 64}. (4) |W|: The size of W is set to max{10, K}. Under this setting, when K ≥ 10, W is aligned with the result list, which enables refilling to be easily executed. (5) efC: efC determines how many nodes are visited in W and plays a role similar to efC in HNSW. It is used to control the trade-off between indexing time and search performance. For fast indexing, efC is recommended to be in [100, 200]. For high graph quality, efC is recommended to be in [1000, 10000]. Notably, thanks to TFB, PAG with a very large efC can be built much faster than HNSW with the same efC.", "source": "marker_v2", "marker_block_id": "/page/7/ListGroup/698"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0089", "section": "5. Experiments", "page_start": 8, "page_end": 8, "type": "Text", "text": "We report main results here. Detailed setup and more results are available in Appendices D and E, respectively.", "source": "marker_v2", "marker_block_id": "/page/7/Text/11"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0090", "section": "5.1. Experimental Setup", "page_start": 8, "page_end": 8, "type": "Text", "text": "Data Statistics. In the main experiments, we evaluate PAG on eight datasets: DBpedia1536 (Qdrant, 2024a) (a.k.a., OpenAI-1536), DBpedia3072 (Qdrant, 2024b) (a.k.a., OpenAI-3072), WoltFood (Qdrant, 2024c) , AmazonBooks (Kang et al., 2024) , DataCompDr (Apple, 2025) ,", "source": "marker_v2", "marker_block_id": "/page/7/Text/13"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0091", "section": "5.1. Experimental Setup", "page_start": 8, "page_end": 8, "type": "Text", "text": "MajorTOM (Major TOM, 2024) , GloVe (Pennington et al., 2014) , and DEEP100M (Simhadri et al., 2021) . The first six are modern text, image, and multimodal (text-to-image) datasets generated by recent embedding models. The latter two are widely-used legacy datasets (downloaded from The Similarity Search Team, CUHK (2018) ). In addition, we report the experimental results on other legacy datasets (Word2Vec, GIST, ImageNet, and SIFT10M) in Appendix E.3. Dataset statistics are reported in Table 2. The queries of DataCompDr are out-of-distribution (OOD). Fig. 11 in Appendix D plots the data and query distributions of this dataset.", "source": "marker_v2", "marker_block_id": "/page/7/Text/14"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0092", "section": "5.1. Experimental Setup", "page_start": 8, "page_end": 8, "type": "Text", "text": "Baselines. Baselines include HNSW (Malkov & Yashunin, 2020) , Vamana (Subramanya et al., 2019) , SymQG (Gou et al., 2025) , ScaNN (Guo et al., 2020a) , IVFPQFS (Jegou ´ et al., 2011) , and RaBitQ+ (Gao et al., 2025) . We design two variants of PAG by tuning its parameters, as shown in Table 4 in Appendix D: (1) PAG-Base, for higher search performance, and (2) PAG-Lite, for faster indexing and smaller memory footprint.", "source": "marker_v2", "marker_block_id": "/page/7/Text/15"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0093", "section": "5.2. Experimental Results", "page_start": 8, "page_end": 8, "type": "Text", "text": "We set the default retrieval size K = 100.", "source": "marker_v2", "marker_block_id": "/page/7/Text/17"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0094", "section": "5.2. Experimental Results", "page_start": 8, "page_end": 8, "type": "ListGroup", "text": "D1. QPS-Recall. Fig. 3 plots the QPS-recall performance. PAG-Base performs the best on all the modern datasets except for high recall settings on WoltFood. Its speedup over HNSW can be up to 5 times. For legacy datasets, PAG-Base is the best on GloVe and only second to SymQG on DEEP100M. PAG-Lite also delivers competitive search performance and is the runner-up on DBpedia1536, DBpedia3072, and MajorTOM. D2. Indexing Time. Fig. 4 (left) shows the indexing time. Under the same or larger efC, PAG-Base requires only 20–40% of the indexing time of HNSW, and is faster than SymQG in most cases. PAG-Lite achieves an indexing time comparable to quantization-based methods, which is further reduced", "source": "marker_v2", "marker_block_id": "/page/7/ListGroup/699"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0095", "section": "5.2. Experimental Results", "page_start": 9, "page_end": 9, "type": "FigureGroup", "text": "Figure 3. QPS-recall, K = 100. SymQG runs out of memory on MajorTOM. Recall is plotted in logarithmic scale to highlight large values.", "source": "marker_v2", "marker_block_id": "/page/8/FigureGroup/347"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0096", "section": "5.2. Experimental Results", "page_start": 9, "page_end": 9, "type": "FigureGroup", "text": "Figure 4. Indexing time and peak memory usage. SymQG runs out of memory on MajorTOM.", "source": "marker_v2", "marker_block_id": "/page/8/FigureGroup/348"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0097", "section": "5.2. Experimental Results", "page_start": 9, "page_end": 9, "type": "Text", "text": "to 0.5 \\times on high-dimensional datasets, where it attains the lowest indexing time.", "source": "marker_v2", "marker_block_id": "/page/8/Text/5"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0098", "section": "5.2. Experimental Results", "page_start": 9, "page_end": 9, "type": "Text", "text": "D3. Memory Footprint. Fig. 4 shows the memory usage in indexing (middle) and searching (right) phases. PAG-Base uses more memory than HNSW on lower-dimensional datasets, but the difference is negligible on higher-dimensional datasets. PAG-Base consistently uses much less memory than SymQG. PAG-Lite achieves the smallest memory footprint in 4 out of 8 cases for both indexing and searching.", "source": "marker_v2", "marker_block_id": "/page/8/Text/6"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0099", "section": "5.2. Experimental Results", "page_start": 9, "page_end": 9, "type": "Text", "text": "D4. High-dimensional Scalability. The competitiveness of PAG-Base and PAG-Lite over d \\in [96,3072] , as shown in Figs. 3 and 4, is consistent, showcasing its insensitivity to dimensionality. In addition, the advantage of PAG in QPS-recall becomes more pronounced on high-dimensional datasets. In contrast, SymQG reports very low recall on high-dimensional datasets DBpedia1536, DBpedia3072, and DataCompDr, where d \\in \\{1536,3072\\} .", "source": "marker_v2", "marker_block_id": "/page/8/Text/7"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0100", "section": "5.2. Experimental Results", "page_start": 9, "page_end": 9, "type": "Text", "text": "\\underline{D5.\\ Retrieval\\ Size\\ Robustness} . Besides the QPS-recall for K=100 in Fig. 3, we report the results for K=10 in Fig. 16 and K=1000 in Fig. 18. PAG methods are highly", "source": "marker_v2", "marker_block_id": "/page/8/Text/8"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0101", "section": "5.2. Experimental Results", "page_start": 9, "page_end": 9, "type": "Text", "text": "competitive across the three K values. When K=10, PAG-Base achieves comparable performance and even outperforms SymQG in the high-recall ( \\geq 95\\% ) region on all datasets except DEEP100M. When K=1000, PAG-Base remains the best, while the performance of SymQG degrades significantly. To highlight the performance comparison, we also plot in Fig. 7 the QPS-recall by varying K from 200 to 800. It can be seen that the gap between PAG-Base and SymQG grows when K moves towards larger values. This observation showcases the robustness of PAG in various applications that differ in retrieval size.", "source": "marker_v2", "marker_block_id": "/page/8/Text/9"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0102", "section": "5.2. Experimental Results", "page_start": 9, "page_end": 9, "type": "Text", "text": "D6. Online Insertion Support. To evaluate the query processing performance with online insertions, we consider the following workload. We randomly sample from the corpus 10,000 vectors as insertion queries and another 10,000 vectors as search queries. The 20,000 vectors are divided into 20 batches, each with 1,000 vectors. The insertion batches and the search batches are interleaved as a workload, with insertion as the first batch. The rest of the corpus is used to build the initial index. Note that DataCompDr is not OOD in this setting because its original query set (in Table 2) is", "source": "marker_v2", "marker_block_id": "/page/8/Text/10"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0103", "section": "5.2. Experimental Results", "page_start": 10, "page_end": 10, "type": "FigureGroup", "text": "Figure 6. QPS-recall, K = 1000.", "source": "marker_v2", "marker_block_id": "/page/9/FigureGroup/353"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0104", "section": "5.2. Experimental Results", "page_start": 10, "page_end": 10, "type": "Text", "text": "not used. To make the processing times of insertion queries and search queries on the same scale, we set efS = efC and tune these two parameters to control the recall. Fig. 8 compares PAG-Base and HNSW, plotting their QPS-recall performances of insertion and search. Insertion queries are slightly slower to process than search queries for both methods. PAG-Base are much faster than HNSW in both insertion and search speeds, and PAG-Base's insertion is even faster than HNSW's search. Similar to PAG-Base's advantage in search, its speedup over HNSW in insertion can be up to 5 times, demonstrating its efficiency in processing online insertions.", "source": "marker_v2", "marker_block_id": "/page/9/Text/3"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0105", "section": "5.2. Experimental Results", "page_start": 10, "page_end": 10, "type": "Text", "text": "Ablation Study. We choose four datasets, DBpedia3072, WoltFood, AmazonBooks, and DataCompDr, for ablation study. Among the six modern datasets, they cover the lowest and highest dimensionality, text and images, as well as OOD. Figs. 9 and 10 show the effectiveness of PAG's components. We observe that TFB consistently reduces indexing time and improves search performance. PES further enhances search performance with negligible additional indexing time. The additional memory usage introduced by TFB and PES is very minor and thus not shown here, because TFB does not affect", "source": "marker_v2", "marker_block_id": "/page/9/Text/4"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0106", "section": "5.2. Experimental Results", "page_start": 10, "page_end": 10, "type": "Text", "text": "the index size and PES barely increases the number of edges.", "source": "marker_v2", "marker_block_id": "/page/9/Text/5"}
+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0107", "section": "6. Conclusion", "page_start": 10, "page_end": 10, "type": "Text", "text": "In this paper, we motivated six critical demands of modern AI applications for ANNS, covering search performance, indexing speed, memory footprint, scalability to dimensionality, robustness against retrieval size, and support of online insertion. To meet these demands, we proposed PAG as a new framework for ANNS. PAG reduces unnecessary distance computations by employing the comparison of exact and approximate distances. The components of PAG are derived from a unified statistical relationship, making its mechanism theoretically explainable. Experiments on modern datasets showcased the superiority of PAG over widely-used ANNS methods as well as state-of-the art solutions, and confirmed the effectiveness of PAG's components.", "source": "marker_v2", "marker_block_id": "/page/9/Text/7"}

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+    {
+      "title": "Approximate Nearest Neighbor Search for Modern AI:\nA Projection-Augmented Graph Approach",
+      "heading_level": null,
+      "page_id": 0,
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+      "title": "Anonymous Authors1",
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+    },
+    {
+      "title": "Abstract",
+      "heading_level": null,
+      "page_id": 0,
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+      ]
+    },
+    {
+      "title": "1. Introduction",
+      "heading_level": null,
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+    {
+      "title": "1.1. Prior Works",
+      "heading_level": null,
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+    },
+    {
+      "title": "1.2. Our Solution",
+      "heading_level": null,
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+      "title": "2. Problem Setting",
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+    {
+      "title": "2.1. Background of RobustPrune",
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+      "title": "2.2. Towards Fast Graph Construction",
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+      "title": "2.3. Towards High Graph Connectivity",
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+      "title": "4. Implementation and Analysis",
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+    {
+      "title": "4.1. Implementation",
+      "heading_level": null,
+      "page_id": 5,
+      "polygon": [
+        [
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+        [
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+    },
+    {
+      "title": "330\n331\n334\n336\n338\n339\n340\n341\n342\n343\n344\n345\n346\n347\n348\n349\n350\n351\n354\n356\n358\n359\n360\n361\n362\n363\n364\n365\nAlgorithm 1: Construction of PAG\nInput: G := \u2205 is the graph to be built; D is the dataset;\n2M is the maximum out-degree; L is the space\npartition size; efC is the maximum size of visited\nnodes; b is the size of W\nOutput: G with node set D, edge set E, and inference\nstructure I\n1 PES set is an empty table to accommodate candidate\nedges;\n2 The capacity of RL is set to efC;\n3 Maximum round Rmax is set to efC/b;\n4 Generate multi-level random-projection structure\nF = {rjl} (1 \u2264 j \u2264 m, 1 \u2264 l \u2264 L);\n5 Select b initial nodes, connect them, and write the\nprojection information w.r.t the connected edges to I. Let\nV denote the set of all initial nodes;\n6 foreach v \u2208 D\\V do\n7 Insert b nodes to W and compute their distances to v;\n8 Compute all \u27e8vl, rjl\u27e9's and build a projection table;\n9 for r = 1 to Rmax do\n10 foreach unvisited u \u2208 W do\n11 foreach w in Nout(u) do\n12 if w passes the PRT-TFB test then\n13 Compute the distance between v\nand w;\n14 if \u2225w \u2212 v\u2225 < \u2225zmax \u2212 v\u2225 then\n15 Update W and insert zmax to\nRT ;\n16 else\n17 Insert w to RF ;\n18 if (u, v) is rejected by the PRT-PES test\nthen\n19 Add (u, v) to the PES set;\n20 Update the nodes in W to RL, and empty W;\n21 Sort and merge RF and RT , and empty RF and\nRT ;\n22 Refill W until it is full using the sorted elements,\nand refill RT with the remaining sorted nodes;\n23 Apply RobustPrune to RL to compute Nout(v) and\nNin(v);",
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+    {
+      "title": "4.2. Algorithm Pseudo-Codes of PAG",
+      "heading_level": null,
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+      "title": "4.3. Complexity Analysis",
+      "heading_level": null,
+      "page_id": 6,
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+    },
+    {
+      "title": "Algorithm 2: Search with PAG",
+      "heading_level": null,
+      "page_id": 6,
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+      "title": "4.4. Parameter Analysis of PAG",
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+    {
+      "title": "5. Experiments",
+      "heading_level": null,
+      "page_id": 7,
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+    {
+      "title": "5.1. Experimental Setup",
+      "heading_level": null,
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+      "title": "5.2. Experimental Results",
+      "heading_level": null,
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+    },
+    {
+      "title": "6. Conclusion",
+      "heading_level": null,
+      "page_id": 9,
+      "polygon": [
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+    },
+    {
+      "title": "References",
+      "heading_level": null,
+      "page_id": 9,
+      "polygon": [
+        [
+          306.0,
+          660.90234375
+        ],
+        [
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+          363.75,
+          670.5
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+        [
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+      ]
+    },
+    {
+      "title": "A. Frequently Used Notations",
+      "heading_level": null,
+      "page_id": 14,
+      "polygon": [
+        [
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+    },
+    {
+      "title": "B. Proof of Theorem 3.1",
+      "heading_level": null,
+      "page_id": 14,
+      "polygon": [
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+        ],
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+      ]
+    },
+    {
+      "title": "C. Related Work",
+      "heading_level": null,
+      "page_id": 15,
+      "polygon": [
+        [
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+    },
+    {
+      "title": "C.1. Discussions on ANNS Solvers",
+      "heading_level": null,
+      "page_id": 16,
+      "polygon": [
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+    {
+      "title": "C.1.1. VECTOR QUANTIZATION",
+      "heading_level": null,
+      "page_id": 16,
+      "polygon": [
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+    },
+    {
+      "title": "C.1.2. SIMILARITY GRAPH",
+      "heading_level": null,
+      "page_id": 16,
+      "polygon": [
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+    },
+    {
+      "title": "C.1.3. QUANTIZED GRAPH",
+      "heading_level": null,
+      "page_id": 16,
+      "polygon": [
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+    },
+    {
+      "title": "C.2. Preliminaries of Routing Test",
+      "heading_level": null,
+      "page_id": 16,
+      "polygon": [
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+    },
+    {
+      "title": "C.2.1. RANDOM PROJECTION FOR ANGLE ESTIMATION",
+      "heading_level": null,
+      "page_id": 16,
+      "polygon": [
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+    },
+    {
+      "title": "C.2.2. ROUTING TEST IN SIMILARITY GRAPHS",
+      "heading_level": null,
+      "page_id": 17,
+      "polygon": [
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+    {
+      "title": "D. Experimental Setup Details",
+      "heading_level": null,
+      "page_id": 17,
+      "polygon": [
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+    },
+    {
+      "title": "D.1. Environment",
+      "heading_level": null,
+      "page_id": 17,
+      "polygon": [
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+    },
+    {
+      "title": "D.2. Baselines",
+      "heading_level": null,
+      "page_id": 17,
+      "polygon": [
+        [
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+        [
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+        [
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+    },
+    {
+      "title": "D.2.1. SELECTION OF BASELINES",
+      "heading_level": null,
+      "page_id": 17,
+      "polygon": [
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+    },
+    {
+      "title": "D.2.2. PARAMETER SETTING",
+      "heading_level": null,
+      "page_id": 17,
+      "polygon": [
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icml26/fddf30e3-e5ae-4a68-b862-daa6e531883a/model_text_v3.txt

@@ -0,0 +1,323 @@
+[p. 1 | section: Abstract | type: Text]
+Approximate Nearest Neighbor Search (ANNS) is fundamental to modern AI applications. Most existing solutions optimize query efficiency but fail to align with the practical requirements of modern workloads. In this paper, we outline six critical demands of modern AI applications: high query efficiency, fast indexing, low memory footprint, scalability to high dimensionality, robustness across varying retrieval sizes, and support for online insertions. To satisfy all these demands, we introduce Projection-Augmented Graph (PAG), a new ANNS framework that integrates projection techniques into a graph index. PAG reduces unnecessary exact distance computations through asymmetric comparisons between exact and approximate distances as guided by projection-based statistical tests. Three key components are designed and unified to the graph index to optimize indexing and searching. Experiments on six modern datasets demonstrate that PAG consistently achieves superior query per second (QPS) recall performance—up to 5×faster than HNSW while offering fast indexing speed and moderate memory footprint. PAG remains robust as dimensionality and retrieval size increase and naturally supports online insertions. Our source code is available at: KejingLu-810/PAG/ .
+
+[p. 1 | section: 1. Introduction | type: Text]
+Given a dataset D ⊂ R d with size n and a query q ∈ R d , Approximate Nearest Neighbor Search (ANNS) aims to find K approximate nearest neighbors of q as accurately and efficiently as possible. Due to its crucial role in many applications such as image search, recommender systems, and
+
+[p. 1 | section: 1. Introduction | type: Text]
+retrieval-augmented-generation (RAG), we have witnessed the rapid proliferation of ANNS solvers. Whereas the ANN-Benchmarks (Bernhardsson et al., 2015) has been developed to compare these methods in a unified environment, many datasets in the ANN-Benchmarks, as pointed out by Chen et al. (2025) , are derived from outdated models such as SIFT and GIST, which are image descriptors developed more than 20 years ago (Lowe, 2004; Oliva & Torralba, 2001) . In addition, its evaluation metric is limited to query per second (QPS)-recall under K = 10.
+
+[p. 1 | section: 1. Introduction | type: Text]
+Seeing the fast development of AI technology, we argue that a well-designed ANNS solver should perform well on modern datasets, and beyond (D1) QPS-recall performance, several demands must also be considered: (D2) indexing time, for which graph indexes such as HNSW (Malkov & Yashunin, 2020) are often criticized (Lin, 2025; Li & Papakonstantinou, 2025) , compromising their use in applications that require instant deployment; (D3) memory footprint, where moderate and adjustable consumption, mostly caused by the index size, is desired for the memory vs accuracy trade-off (Cheng et al., 2024; Wallace, 2025) ; (D4) scalability to high dimensionality, as motivated by the increasing dimensionality of modern embedding models such as CLIP (Radford et al., 2021) 1 ; (D5) robustness against the retrieval size K, due to the needs in various applications (e.g., K is typically 10 for RAG (Fensore et al., 2025; Ke et al., 2025) but can be up to hundreds in image retrieval (Vendrow et al., 2024) and thousands in recommender systems (Zhang et al., 2025) ); (D6) support of online insertions, which is essential for emergent applications such as self-evolving agents that continually accumulate and reuse experience through interaction (Ouyang et al., 2025; Zhai et al., 2025; Zhang et al., 2026) .
+
+[p. 1 | section: 1.1. Prior Works | type: Text]
+We review representative ANNS solvers based on D1 – D6. More discussions on literature can be found in Appendix C.
+
+[p. 1 | section: 1.1. Prior Works | type: Footnote]
+1 Despite the availability of dimensionality reduction techniques, Weller et al. (2025) proved that for embedding-based retrieval with dimensionality d, there exists a dataset size n (e.g., n ≈ 4M for d = 1024) beyond which it is theoretically impossible to represent all possible top-2 document combinations, showcasing the necessity for higher dimensionality on large datasets.
+
+[p. 2 | section: 1.1. Prior Works | type: TableGroup]
+Table 1. Summary of empirical performance comparison of notable ANNS solvers, HNSW (Malkov & Yashunin, 2020), Vamana (in-memory DiskANN (Subramanya et al., 2019)), IVFPQFS (fast scan IVFPQ (Jégou et al., 2011)), ScaNN (Guo et al., 2020a), RaBitQ+ (Gao et al., 2025), SymQG (Gou et al., 2025), HNSW+KS2 (Lu et al., 2025), and our solutions PAG-Base (for high QPS) and PAG-Lite (for fast indexing and small index size). Detailed results are available in Sec. 5.2. The evaluation of D1 – D3 is clustered into tiers, where Tier-4 is the best. D4 – D6 are evaluated in yes/no (√/X). Graph-Based Quantization-Based QG PG PA ıG Criteria HNSW Vamana IVFPQFS 1 ScaNN 1 RaBitQ+ SymQG HNSW+KS2 PAG-Base PAG-Lite D1. QPS-recall Tier-2 Tier-2 Tier-1 Tier-1 Tier-1 Tier-3 Tier-3 Tier-4 Tier-3 D2. Indexing time Tier-1 Tier-1 Tier-3 Tier-3 Tier-3 Tier-2 Tier-1 Tier-2 Tier-4 D3. Memory footprint 2 Tier-3 Tier-3 Tier-4 Tier-4 Tier-4 Tier-1 Tier-2 Tier-3 Tier-4 D4. High-dim. scalability ✓ ✓ ✓ ✓ ✓ X ✓ ✓ ✓ D5. Retrieval size robustness ✓ ✓ ✓ ✓ ✓ Х ✓ ✓ ✓ D6. Online insertion support 3 ✓ ✓ ✓ ✓ \checkmark X × ✓ ✓ Reranking is enabled for IVFPOFS and ScaNN.
+
+[p. 2 | section: 1.1. Prior Works | type: Text]
+Graph-Based Methods. These methods construct a similarity graph connecting nearby vectors and hop towards the neighborhood of q. Notable methods are HNSW (Malkov & Yashunin, 2020), NSG (Fu et al., 2019), and DiskANN (Subramanya et al., 2019). They are generally competitive in QPS-recall but slow in building indexes.
+
+[p. 2 | section: 1.1. Prior Works | type: Text]
+Quantization-Based Methods. These methods compress vectors and rank them using approximate (i.e., quantized) distances to q in pursuit of efficiency. Representative methods are IVFPQ (Jégou et al., 2011) and ScaNN (Guo et al., 2020a). They are fast in building indexes and save memory, but their QPS-recall is generally inferior to graph-based methods.
+
+[p. 2 | section: 1.1. Prior Works | type: Text]
+Projection-Based Methods. These methods use random (e.g., E2LSH (Andoni & Indyk, 2005), Falconn (Andoni et al., 2015), and CEOs (Pham, 2021)) or data-dependent (e.g., learning to hash (Wang et al., 2018)) projections of the original vectors for indexing. Although many of them enjoy theoretical guarantees, they are less competitive than graphand quantization-based methods in QPS-recall performance, hence becoming less popular in modern top-K ANNS.
+
+[p. 2 | section: 1.1. Prior Works | type: Text]
+<u>Tree-Based Methods.</u> They utilize tree indexes. table methods include k-d tree (Bentley, 1990), cover tree (Beygelzimer et al., 2006), and Annoy (Bernhardsson, 2013). Like projection-based methods, they are less widely used for modern ANNS due to limited QPS-recall performance.
+
+[p. 2 | section: 1.1. Prior Works | type: Text]
+As discussed above, graph-based methods are good at QPSrecall. But they compute exact distances between vectors, which is computationally costly. To address this weakness, recent advancements attempt to integrate other techniques, in particular, quantization or projection, into graphs.
+
+[p. 2 | section: 1.1. Prior Works | type: Text]
+Quantized Graph (QG) Methods. They construct a similarity graph based on quantized vectors instead of the original ones, thereby replacing exact distances with approximate values. Representative methods are NGT-QG (Yahoo! Japan, 2023),
+
+[p. 2 | section: 1.1. Prior Works | type: Text]
+LVQ (Aguerrebere et al., 2023), and SymphonyQG (Gou et al., 2025). QG methods can achieve very high QPS-recall performance, yet they are sensitive to the data distribution and many of them do not perform very well in D3 - D6.
+
+[p. 2 | section: 1.1. Prior Works | type: Text]
+Projection + Graph ( PG ) Methods . Unlike QG, they operate on original vectors and reduce unnecessary exact distance computations. To this end, they employ projection techniques to test whether a neighbor needs to be explored (called routing tests). Representative methods are FINGER (Chen et al., 2023), PEOs (Lu et al., 2024), and KS2 (Lu et al., 2025). Despite QPS-recall improvement on top of graph-based methods, such improvement comes at the cost of indexing time, memory footprint, and support for online insertions.
+
+[p. 2 | section: 1.1. Prior Works | type: Text]
+Table 1 compares notable methods implemented in leading vector databases and recent ones representing state of the art.
+
+[p. 2 | section: 1.2. Our Solution | type: Text]
+We propose a new ANNS framework, Projection-Augmented Graph (PAG), which integrates projection into a similarity graph and achieves superior performance for all the six criteria. Unlike PG methods, PAG treats projection as a fundamental building block of graph construction rather than as a plug-in. The rationale of PAG is to accommodate both exact and approximate distances within a unified framework, addressing two key issues: (1) when distances shall be computed exactly and when they shall be approximated, and (2) how exact and approximate distances are compared during indexing and searching to enhance the performance.
+
+[p. 2 | section: 1.2. Our Solution | type: Text]
+PAG significantly reduces the computational cost of searching (D1) and indexing (D2) by carefully determining whether exact distance computation is needed. Such procedure relies on asymmetric comparisons between exact and approximate distance values obtained from space-efficient and adjustable random-projection structures (D3), as opposed to symmetric comparisons in QG. Moreover, the asymmetric distance
+
+[p. 2 | section: 1.2. Our Solution | type: Footnote]
+Memory footprint for searching, rather than indexing, is compared.
+
+[p. 2 | section: 1.2. Our Solution | type: Footnote]
+& lt;sup>3</sup> For any streaming workload Q composed of search and insertion queries, we say an algorithm supports online insertion, if the amortized cost of processing a query in Q is O(1) times the cost of a search. In other words, the index can be incrementally updated with minimal cost. See Sec. 4.3 for the analysis of PAG.
+
+[p. 3 | section: 1.2. Our Solution | type: Text]
+comparisons, as guided by random-projection-based statistical tests, only need to tell which value is larger, thereby achieving accuracy while preserving efficiency. Meanwhile, PAG inherits graph-based methods' scalability to high dimensionality (D4) and robustness against retrieval size (D5), and follows the search-and-insertion paradigm (e.g., HNSW) to accommodate online insertions (D6).
+
+[p. 3 | section: 1.2. Our Solution | type: Text]
+Our technical contributions are summarized as follows.
+
+[p. 3 | section: 1.2. Our Solution | type: ListGroup]
+(1) Following the idea of routing tests in PEOs (Lu et al., 2024) and KS2 (Lu et al., 2025) , we derive a Probabilistic Routing Test (PRT) function (Sec. 3.2) . The theoretical result (Theorem 3.1) , along with Lemma 4.3 in Lu et al. (2025) , provides a complete theoretical explanation of PRT. In addition, our work is the first to apply PRT to graph construction. (2) We propose a data structure called Test Feedback Buffer (TFB) (Sec. 3.3) , which refines the threshold setting in PRT and enables the reuse of false positives generated by PRT. By incorporating TFB into PRT, we obtain the PRT-TFB test as a core technique for accelerating both indexing and searching. (3) We propose a statistical test called Probabilistic Edge Selection (PES) (Sec. 3.4) , which is derived from Theorem 3.1 and can expand in-degrees when necessary. In collaboration with PRT, PRT-PES can improve the search performance for hard datasets on which traditional graph indexes perform poorly, while incurring very small indexing overhead. (4) PRT, TFB, and PES constitute PAG (Fig. 1) . We show how the three components interact within PAG, with implementation details and complexity analysis provided. (5) We conduct experiments on six modern (post-2023) datasets covering text, image, and multimodal data, with dimensionality ranging from 384 to 3072 and retrieval size from 10 to 1000. The results show that PAG achieves the best QPS-recall performance (up to 5× faster than HNSW), and its superiority is particularly evident on datasets with higher dimensionality. PAG remains dominant as K increases. By adjusting parameters, PAG can deliver the fastest indexing speed and lowest memory footprint on most datasets while maintaining competitive QPS-recall. PAG also performs well on the datasets in the ANN-Benchmarks, showcasing its compatibility with data generated by legacy models.
+
+[p. 3 | section: 2. Problem Setting | type: Text]
+Since PAG adopts a search-and-insertion paradigm, its search process, similar to HNSW, can be seen as a special case of index construction (i.e., no update to the graph), and we can focus on the problems that arise in index construction. Because constructing a similarity graph is essentially determining its edge set, we consider the following question: Can we design statistical methods that select edges efficiently (i.e., fast indexing) and effectively (i.e., fast searching)? To an-
+
+[p. 3 | section: 2. Problem Setting | type: Text]
+swer this question, we first review the existing edge-selection strategy, and then formalize the problems to be solved.
+
+[p. 3 | section: 2.1. Background of RobustPrune | type: Text]
+Given a directed graph, let v be a node whose edges are to be determined. To identify its in-neighbor set Nin(v) and out-neighbor set Nout(v), state-of-the-art methods, such as HNSW, NSG and Vamana, employ a strategy known as RobustPrune to decide whether a candidate edge should be preserved. Take the construction of Nin(v) as an example. For a candidate neighbor u of node v, the pruning criterion considers the following set 2 :
+
+[p. 3 | section: 2.1. Background of RobustPrune | type: Equation]
+S_{u,v} = \{ w \in N_{\text{out}}(u) \mid ||w - v|| \le ||v - u|| \}. (1)
+
+[p. 3 | section: 2.1. Background of RobustPrune | type: Text]
+In RobustPrune, if there exists w in Su, v such that ∥w − u∥ ≤ ∥v − u∥, u is not added to Nin(v); otherwise, Nin(v) is updated as Nin(v)∪{u}. This criterion admits an intuitive interpretation: if ∥w − u∥ ≤ ∥v − u∥ holds for some w, it is likely that one can reach v from u via w along a monotonic search path (Fu et al., 2019) , implying that the direct edge uv⃗ is redundant. Similarly, Nout(v) can be constructed by RobustPrune with the roles of u and v reversed. In sum, existing graph-based methods determine Nout(v) and Nin(v) sequentially in the following manner.
+
+[p. 3 | section: 2.1. Background of RobustPrune | type: Text]
+Nout(v) . We take v as the query and conduct an ANNS from v to obtain a candidate set, which is stored in a priority queue P. RobustPrune is then applied to P to determine Nout(v).
+
+[p. 3 | section: 2.1. Background of RobustPrune | type: Text]
+Nin(v) . We check every u in Nout(v) and preserve those u's retained by RobustPrune as Nin(v).
+
+[p. 3 | section: 2.2. Towards Fast Graph Construction | type: Text]
+During graph construction, most of the time is spent on ANNS for the candidate set of Nout(v). Although we can reduce the graph construction parameter (e.g., efC in HNSW) for fast graph construction, the query processing performance degrades accordingly. Thus, we raise the following question.
+
+[p. 3 | section: 2.2. Towards Fast Graph Construction | type: Text]
+Q1. How can we accelerate the ANNS for v while preserving the query processing performance?
+
+[p. 3 | section: 2.2. Towards Fast Graph Construction | type: Text]
+Existing studies PEOs and KS2 have shown that PRT has the potential to be an appropriate answer to Q1. For each visited node u and its out-neighbor w, PRT sends (u, v, w) to a probabilistic test function to see whether they can pass the test, and then determines whether the exact distance between v and w needs to be computed based on the test result. Consequently, PRT avoids unnecessary distance computations, leading to faster search. In KS2 (Lu et al., 2025) , it is shown that the routing test under ℓ 2 and cosine distances is equivalent to the angle-thresholding problem when vector norms
+
+[p. 3 | section: 2.2. Towards Fast Graph Construction | type: Footnote]
+For Vamana, we assume its pruning ratio α = 1 and omit it in the inequality.
+
+[p. 4 | section: 2.2. Towards Fast Graph Construction | type: FigureGroup]
+Figure 1. An overview of PAG. In this example, \boldsymbol{u} has 7 out-neighbors \{\boldsymbol{w_i}\}_{i=1}^7 . Let \{\boldsymbol{e_i}\}_{i=1}^7 denote the edges between \boldsymbol{u} and \{\boldsymbol{w_i}\}_{i=1}^7 , which are sent to PRT, where the threshold is determined by \boldsymbol{z_{max}} . As a result, only \boldsymbol{w_3} and \boldsymbol{w_5} pass PRT, and their exact distances to \boldsymbol{v} are computed. By distance comparison, node \boldsymbol{w_5} is identified as a false positive, not added to W, and thus sent to R_F . Node \boldsymbol{w_3} is inserted into W, causing \boldsymbol{z_{max}} to be ejected from W and sent to R_T . The two rings R_F and R_T are merged and refilled into W. On the other route, by a left shift, the signs of both \boldsymbol{w_3} and \boldsymbol{w_5} are reversed, and all signs become negative, indicating that \boldsymbol{u} is rejected by the PES test. Consequently, \boldsymbol{u}\dot{\boldsymbol{v}} is treated as a candidate edge and added to the PES set.
+
+[p. 4 | section: 2.2. Towards Fast Graph Construction | type: Text]
+are stored explicitly. Therefore, PRT can be formulated as follows:
+
+[p. 4 | section: 2.2. Towards Fast Graph Construction | type: Text]
+Problem 2.1. ( Probabilistic Routing Test ) Given u, v, w \in \mathbb{S}^{d-1} and a threshold -1 \le \tau \le 1 , construct a random-projection vector set \mathcal{F} and design a test function w.r.t \mathcal{F} , i.e., PRT : (u, v, w, \tau) \to RV with time complexity o(d), where RV denotes the set of one-dimensional random variables, such that if \cos(\angle(v-u, w-u)) \ge \tau , then \mathbb{P}[\operatorname{PRT}(u, v, w, \tau) \ge 0] \ge 0.5 . Otherwise, \mathbb{P}[\operatorname{PRT}(u, v, w, \tau) < 0] \to 1 as |\mathcal{F}| \to \infty .
+
+[p. 4 | section: 2.2. Towards Fast Graph Construction | type: Text]
+Different from the routing test functions used in PEOs and KS2, we take \tau as an explicit input. In this paper, we use another method to determine \tau in order to further improve the efficiency of the routing test.
+
+[p. 4 | section: 2.3. Towards High Graph Connectivity | type: Text]
+From the process of determining N_{\rm in}(\boldsymbol{v}) , we observe that its candidate set is restricted to N_{\rm out}(\boldsymbol{v}) . Recent work (Wang et al., 2025a) shows that |N_{\rm out}(\boldsymbol{v})| may be too small to reliably determine incoming edges, causing some nodes to have very small in-degrees. Such nodes may become unreachable during search, which partly explains why HNSW performs poorly on certain real-world datasets. On the other hand, for each checked node \boldsymbol{u} , the worst-case time complexity of RobustPrune can reach O(|N_{\rm out}(\boldsymbol{u})|d) , implying that a straightforward enlargement of the candidate set for
+
+[p. 4 | section: 2.3. Towards High Graph Connectivity | type: FigureGroup]
+Figure 2. Illustration of PES (left: the role of PES; right: geometric illustration). Let \boldsymbol{v} be the node to be inserted. The four green nodes are out-neighbors of \boldsymbol{v} . Without PES, we can obtain only a single in-neighbor via RobustPrune. By taking all other visited nodes (the blue ones) into account, we apply PES, followed by RobustPrune. As such, we can identify three additional promising in-neighbors, thereby strengthening the connectivity of the neighborhood of \boldsymbol{v} .
+
+[p. 4 | section: 2.3. Towards High Graph Connectivity | type: Text]
+determining incoming edges is expensive. Considering this dilemma, we raise the following question.
+
+[p. 4 | section: 2.3. Towards High Graph Connectivity | type: Text]
+Q2. How can we efficiently detect incoming edges outside N_{\text{out}}(v) that are useful for improving query processing performance but hard to be identified by RobustPrune?
+
+[p. 5 | section: 2.3. Towards High Graph Connectivity | type: Text]
+To answer this question, we formulate a problem as follows.
+
+[p. 5 | section: 2.3. Towards High Graph Connectivity | type: Text]
+Problem 2.2. ( Probabilistic Edge Selection ) Given u, v \in \mathbb{S}^{d-1} , construct a random-projection-vector set \mathcal{F} and design a probabilistic edge selection function PES: (u, v) \to RV whose time complexity is O(|N_{\mathrm{out}}(u)|) , such that if S_{u,v} is non-empty, then \mathbb{P}[\mathrm{PES}(u,v) \geq 0] \geq 0.5 . Otherwise, \mathbb{P}[\mathrm{PES}(u,v) < 0] \to 1 as |\mathcal{F}| \to \infty .
+
+[p. 5 | section: 2.3. Towards High Graph Connectivity | type: Text]
+We emphasize that the PES test is far more than a fast probabilistic implementation of RobustPrune. This is because the PES test can be applied to all visited nodes during the ANNS of \boldsymbol{v} , whose number is typically much larger than N_{\rm out}(\boldsymbol{v}) in practice, leading to a better graph index in terms of connectivity (Fig. 2).
+
+[p. 5 | section: 3. Projection-Augmented Graph | type: Text]
+We show that Problems 2.1 and 2.2 can be solved in a unified framework PAG, which contains three components: Probabilistic Routing Test (PRT), Test Feedback Buffer (TFB), and Probabilistic Edge Selection (PES).
+
+[p. 5 | section: 3.1. Neighborhood Relations via Random Projection | type: Text]
+We present an asymptotic result that characterizes the relationship between multiple angles in high-dimensional spaces and their corresponding projection values onto a certain projection vector. This result forms the theoretical basis for all the three components. We first describe how to construct the random-projection vector set \mathcal F that appears in Problems 2.1 and 2.2, following an approach similar to that used in KS2 (Lu et al., 2025). Specifically, we divide the original space \mathbb R^d into L subspaces, each of dimension d/L. In each subspace, we generate multiple cross-polytopes and apply an independent rotation to each, producing a total of m normalized vectors on \mathbb S^{d/L-1} . By concatenation, we obtain a total of m^L normalized vectors \{r_j\}_{j=1}^{m^L} , which together form the set \mathcal F . Consider v as the node to be inserted and v as the candidate neighbor to be checked, let N_{\mathrm{out}}(u) := \{w_i\}_{i=1}^t . We define \{\alpha_i\}_{i=1}^t as follows.
+
+[p. 5 | section: 3.1. Neighborhood Relations via Random Projection | type: Equation]
+\alpha_i := \arccos \frac{\langle \boldsymbol{w_i} - \boldsymbol{u}, \boldsymbol{v} - \boldsymbol{u} \rangle}{\|\boldsymbol{w_i} - \boldsymbol{u}\| \|\boldsymbol{v} - \boldsymbol{u}\|}, \quad 1 \le i \le t \quad (2)
+
+[p. 5 | section: 3.1. Neighborhood Relations via Random Projection | type: Text]
+Since \|\boldsymbol{w_i} - \boldsymbol{u}\| and \|\boldsymbol{v} - \boldsymbol{u}\| can be pre-computed, we aim to estimate [\cos \alpha_1, \dots, \cos \alpha_t]^{\top} without exact inner product computation in Eq. (2). We will show that this can be realized by \mathcal{F} . For each \boldsymbol{w_i} ( 1 \leq i \leq t ), let \boldsymbol{r_i^*} \in \mathcal{F} be the reference vector that has the smallest angle with \boldsymbol{w_i} - \boldsymbol{u} , and denote this smallest angle by \beta_i . We use \cos \theta_i to denote the cosine of the angle between \boldsymbol{r_i^*} and \boldsymbol{v} - \boldsymbol{u} , and subscript l to denote the l-th sub-vector of the original vector in \mathbb{R}^d , 1 \leq l \leq L . We introduce the following assumptions for each i:
+
+[p. 5 | section: 3.1. Neighborhood Relations via Random Projection | type: Equation]
+(A1) \|(\boldsymbol{w_i} - \boldsymbol{u})\| = 1 and \|(\boldsymbol{v} - \boldsymbol{u})\| = 1 .
+
+[p. 5 | section: 3.1. Neighborhood Relations via Random Projection | type: Equation]
+(A2) \|(\boldsymbol{w_i} - \boldsymbol{u})_l\| and \|(\boldsymbol{v} - \boldsymbol{u})_l\| are equal to (1 + o(1))/\sqrt{L} . (A3) \langle (\boldsymbol{v} - \boldsymbol{u})_l, (\boldsymbol{w_i} - \boldsymbol{u})_l \rangle = (\cos \alpha_i)(1 + o(1))/L .
+
+[p. 5 | section: 3.1. Neighborhood Relations via Random Projection | type: Text]
+Here, A1 does not lose generality, and A2 and A3 are mild for large d and L (see the remarks in Appendix B). Under these assumptions, the following theorem shows that \{\alpha_i\}_{i=1}^t can be asymptotically estimated by \{\beta_i\}_{i=1}^t and \{\theta_i\}_{i=1}^t .
+
+[p. 5 | section: 3.1. Neighborhood Relations via Random Projection | type: Text]
+Theorem 3.1. Under A1–A3, as L \to \infty , d/L \to \infty , and m grows sufficiently fast with respect to d/L, X = [\cos \theta_1, \dots, \cos \theta_t]^\top , conditioned on \{\alpha_i, \beta_i\}_{i=1}^t , is asymptotically Gaussian:
+
+[p. 5 | section: 3.1. Neighborhood Relations via Random Projection | type: Equation]
+X \mid \{\alpha_i, \beta_i\}_{i=1}^t \xrightarrow{d} \mathcal{N}(\bar{\mu}, \bar{\Sigma}_{m,L}) (3)
+
+[p. 5 | section: 3.1. Neighborhood Relations via Random Projection | type: Text]
+where the mean is \bar{\mu} = [\cos \alpha_1 \cos \beta_1, \dots, \cos \alpha_t \cos \beta_t]^\top , and \bar{\Sigma}_{m,L} = O(\epsilon_m/L) with \epsilon_m \to 0 as m \to \infty .
+
+[p. 5 | section: 3.1. Neighborhood Relations via Random Projection | type: Text]
+Remarks. (1) Theorem 3.1 establishes a geometric relationship among multiple out-neighbors of \boldsymbol{u} and \boldsymbol{v} , allowing us to estimate the relative location of \boldsymbol{v} w.r.t. \boldsymbol{u} within the neighborhood of \boldsymbol{u} . This relationship is the key to both PRT and PES. (2) In our proof, m \to \infty is used to eliminate noise from off-diagonal entries, while L \to \infty is required for the central limit theorem (CLT). In practice, moderate values of m and L are sufficient. Weak results under finite (m, L) can be found in Lemma 4.3 in the KS2 paper (Lu et al., 2025). (3) \boldsymbol{X} can be computed efficiently by AVX512 on modern CPUs. Vector [\cos \beta_1, \ldots, \cos \beta_t]^{\top} can be pre-computed, meaning that we can obtain an efficient way to estimate all \cos \alpha_i 's simultaneously.
+
+[p. 5 | section: 3.2. Probabilistic Routing Test | type: Text]
+Let \tau_i be a threshold w.r.t w_i . Our PRT function is as follows:
+
+[p. 5 | section: 3.2. Probabilistic Routing Test | type: Equation]
+PRT(\boldsymbol{u}, \boldsymbol{v}, \boldsymbol{w_i}, \tau_i) = \frac{\cos \theta_i}{\cos \beta_i} - \tau_i. (4)
+
+[p. 5 | section: 3.2. Probabilistic Routing Test | type: Text]
+For fixed (u, v), if the value of PRT is positive, the corresponding w_i passes the PRT. Based on Theorem 3.1, under asymptotic assumptions, the PRT function satisfies all the required properties in Problem 2.1. Here, we do not specify the setting of \tau_i , which will be postponed to Sec. 3.3.
+
+[p. 5 | section: 3.2. Probabilistic Routing Test | type: Text]
+Remarks. Following the use of routing test in PEOs and KS2, exact distances are computed only between v and those w_i 's that pass PRT (see Fig. 1). The PRT function has the same structure as the KS2 test function (Lu et al., 2025), except for the setting of the threshold \tau . However, since the PRT and KS2 tests are derived using different principles, their theoretical results are complementary. Theorem 4.3 in the KS2 paper shows that, for every single w_i , even for finite (m, L), if \cos(\angle(v - u, w_i - u)) \ge \tau_i , then \mathbb{P}[\operatorname{PRT}(u, v, w_i, \tau_i) \ge 0] \ge 0.5 still holds. Our result, on the other hand, characterizes the concrete asymptotic distribution and reveals the impact of L on the covariance, thereby explaining how L is used to adjust the estimation accuracy.
+
+[p. 6 | section: 3.3. Test Feedback Buffer | type: Text]
+For the PRT function, a key issue is how to set an appropriate threshold τ . In PEOs and KS2, τ is simply determined by the current furthest point in the result list (priority queue) P. However, due to the existence of false positives (FPs) generated by PRT, some points may pass PRT but cannot be added to P, which implies that the exact distance computations w.r.t such points are redundant. On the other hand, the Gaussian distribution established in Theorem 3.1 implies that, with high probability, the actual distances of these FPs to the query q are not much larger than the threshold τ . This observation naturally raises the following question: can we incrementally increase the threshold τ so that the currently generated FPs can be reused in subsequent search process?
+
+[p. 6 | section: 3.3. Test Feedback Buffer | type: Text]
+We give an affirmative answer to this question and propose TFB, which consists of four components: the top-K result list RL, the working set W, and two ring buffers R F and R T , where |W| = |R F | = |R T |. The search procedure is divided into multiple rounds. In the j-th round, we operate on W and adopt the standard Best-First Search (BFS) strategy to update its nodes. During the BFS over W, in addition to the nodes stored in W, there are two types of nodes whose exact distances to the query q are computed. The first type consists of nodes ejected from W; these nodes are inserted into R T . The second type consists of false-positive (FP) nodes that pass the PRT but cannot be added to W; these nodes are inserted into R F . After all nodes in W have been visited, we update R L and clear W. We then merge R F and R T and sort the combined list by distance to q. The sorted nodes are first inserted into W until it is full, and the remaining nodes are inserted into R T . Finally, R F is cleared. The ANNS is completed after several such rounds. In the j-th round, let zmax denote the furthest node in W. τ i in Eq. (4) is set to be the threshold on the cosine of the angle between w i − u and v − u required for w i to be admitted into W, i.e.,
+
+[p. 6 | section: 3.3. Test Feedback Buffer | type: Equation]
+\tau_i = \frac{\|u - w_i\|^2 + \|u - v\|^2 - \|z_{\max} - v\|^2}{2\|v - u\|\|w_i - u\|}. (5)
+
+[p. 6 | section: 3.3. Test Feedback Buffer | type: Text]
+PRT-TFB Test. We call the combination of Eq. (4) and Eq. (5) the PRT-TFB test. Compared with existing routing test methods such as PEOs and KS2, which operate on the whole priority queue P, the PRT-TFB test has the following two advantages. (1) For large efC or large efS—which denotes the result list size, same as the notation in HNSW—the size of W is much smaller than that of P, and the movement of elements in W is much faster than that in P. (2) Each FP has a good chance of being selected as the visited node in the future rounds only if it remains in either of two rings, ensuring that its exact distance to q can be utilized at a certain time point.
+
+[p. 6 | section: 3.4. Probabilistic Edge Selection | type: Text]
+In RobustPrune, ∥w i − v∥ needs to be smaller than ∥v − u∥ to allow u to be retained. Let δ i := ∥w i − u∥/(2∥v − u∥) denote the threshold on the cosine of the angle between w i − u and v − u, such that ∥w i − v∥ < ∥v − u∥. Then, the PES function is designed as follows.
+
+[p. 6 | section: 3.4. Probabilistic Edge Selection | type: Equation]
+PES(\boldsymbol{u}, \boldsymbol{v}) = \max_{1 \le i \le t} \left( \frac{\cos \theta_i}{\cos \beta_i} - \delta_i \right). (6)
+
+[p. 6 | section: 3.4. Probabilistic Edge Selection | type: Text]
+By Theorem 3.1, we can see that the PES function in Eq. (6) satisfies the probabilistic properties in Problem 2.2. In practice, if the value in Eq. (6) is negative, we say that (u, v) is rejected by PES, and this edge will be regarded as a promising candidate for further examination of RobustPrune.
+
+[p. 6 | section: 3.4. Probabilistic Edge Selection | type: Text]
+PRT-PES Collaboration. From their definitions, the difference between the two thresholds, that is, τ i − δ i , is a value independent of v − w and can be pre-computed. Thus, we can conduct the PES test based on the test results of the PRT, leading to an O(1) time complexity, which confirms the objective O(|Nout(u)|) stated in Problem 2.2.
+
+[p. 6 | section: 3.4. Probabilistic Edge Selection | type: Text]
+PES Set. For every u whose PES value is negative, we do not check uv⃗ by RobustPrune immediately. Instead, we add uv⃗ into a so-called PES set for later examination (see Fig. 1) .
+
+[p. 6 | section: 4.1. Implementation | type: Text]
+Indexing Phase. PAG inserts every v ∈ D sequentially into the graph. For each v, the process can be divided into the following three steps.
+
+[p. 6 | section: 4.1. Implementation | type: Text]
+Step 1. PAG performs ANNS equipped with the PRT-TFB test to obtain the elements stored in RL, and then uses RobustPrune to determine Nout(v) and Nin(v).
+
+[p. 6 | section: 4.1. Implementation | type: Text]
+Step 2. Let Cand(v) denote the set of all nodes visited during ANNS. PAG uses PRT-PES to select additional in-neighbors from Cand(v), thereby supplementing the PES set.
+
+[p. 6 | section: 4.1. Implementation | type: Text]
+Based on our preceding discussion, PRT-TFB ensures the efficiency of ANNS, while PRT-PES detects more promising edges. We finally select more edges from the PES set.
+
+[p. 6 | section: 4.1. Implementation | type: Text]
+Step 3. For each candidate edge in the PES set, we apply RobustPrune to determine if it can be added to the graph.
+
+[p. 6 | section: 4.1. Implementation | type: Text]
+Searching Phase. The query processing with PAG is exactly the PRT-TFB-based ANNS for a query q.
+
+[p. 6 | section: 4.1. Implementation | type: Text]
+Online Insertion. Like HNSW, PAG naturally supports online insertion. The PES set is checked only after enough new nodes have been inserted into the graph.
+
+[p. 7 | section: 4.2. Algorithm Pseudo-Codes of PAG | type: Text]
+The pseudo-codes of the PAG algorithms are presented in Algs. 1 and 2, which represent the indexing and searching, respectively. Note that indexing is essentially the sequential insertion of all nodes. Similar to Lu et al. (2024) , we store the computed projections in a projection table for lookup (Line 8, Alg. 1 and Line 4, Alg. 2) .
+
+[p. 7 | section: 4.2. Algorithm Pseudo-Codes of PAG | type: Text]
+24 Add the projection information of new edges to I; 25 Apply RobustPrune to the PES set, supplement Nin(v)'s, and add the projection information of new edges to I;
+
+[p. 7 | section: 4.3. Complexity Analysis | type: Text]
+Time complexity. Searching with HNSW is in O(n ′d)-time, where n ′ is the number of visited nodes during the search. Although a rigorous analysis of n ′ remains an open problem, n ′ can be roughly estimated as O(M log n), where 2M is the
+
+[p. 7 | section: Algorithm 2: Search with PAG | type: Code]
+Input: G(D, E, I) is the constructed similarity graph; F is the random-projection structure; efS is the result list size; q is the query; K is the retrieval size Output: Top-K ANN nodes 1 The capacity of RL is set to K; 2 b = max{10, K}; 3 Maximum round Rmax is set to efS/b; 4 Compute all ⟨ql, rjl⟩'s and build a projection table; 5 Insert b elements to W and compute their distance to q; 6 for r = 1 to Rmax do 7 foreach unvisited u ∈ W do 8 foreach w in Nout(u) do 9 if w passes PRT-TFB test then 10 Compute the distance between q and w; 11 if ∥w − q∥ < ∥zmax − q∥ then 12 Update W and insert zmax to RT ; 13 else 14 Insert w to RF ; 15 Update the nodes in W to RL, and empty W; 16 Sort and merge RF and RT , and empty RF and RT ; 17 Refill W until it is full using the sorted elements, and refill RT with the remaining sorted nodes; 18 Return the top-K nodes in RL;
+
+[p. 7 | section: Algorithm 2: Search with PAG | type: Text]
+maximum out-degree. In contrast, the complexity of PAG is O(n ′L+γn′d), whereL ≪ d is the user-specified parameter and γ ≤ 0.5 denotes the ratio of nodes that passed the PRT-PES test relative to the total number of candidates checked. γ is small in practice (generally < 0.2), and is smaller than the ratio in PEOs and KS2, because TFB ensures that the threshold increases incrementally rather than being set to the furthest distance w.r.t. a full priority queue.
+
+[p. 7 | section: Algorithm 2: Search with PAG | type: Text]
+As for indexing, due to the search-and-insertion paradigm, we focus on the insertion of v, for which the complexity is O(n ′L + γn′d + M2d + M dm). Here, O(M2d) is the complexity of node connection, and O(M dm) is used to compute projection information, both of which are much smaller than the ANNS part due to their independence of n ′ .
+
+[p. 7 | section: Algorithm 2: Search with PAG | type: Text]
+Space complexity. The space complexity of searching with PAG is O(nd + nM + nML), where O(nd) corresponds to the original dataset, O(nM) corresponds to the edge set and O(nML) corresponds to the inference structure. In the indexing phase, since we adopt in-place search for insertion, we only require an additional space of at most O(Mn) to store the edges in PES set. To further reduce the memory footprint, we can follow LVQ (Aguerrebere et al., 2023) and represent floats with 2 bytes.
+
+[p. 7 | section: 4.4. Parameter Analysis of PAG | type: Text]
+We analyze the settings of the parameters in PAG.
+
+[p. 7 | section: 4.4. Parameter Analysis of PAG | type: Text]
+(1) m: The value of m is fixed to 16, so that each projection vector in a subspace is represented by 4 bits, which is com-
+
+[p. 8 | section: 4.4. Parameter Analysis of PAG | type: Text]
+Table 2. Dataset statistics.
+
+[p. 8 | section: 4.4. Parameter Analysis of PAG | type: Text]
+Name/Source Dataset Size Query Size Dim. OOD Type Embedding Model Distance Measure Modern Datasets DBpedia1536 999,000 1,000 1,536 No Text text-embedding-3-large (OpenAI, 2024) Euclidean DBpedia3072 999,000 1,000 3,072 No Text text-embedding-3-large (OpenAI, 2024) Euclidean WoltFood 1,719,611 1,000 512 No Image clip-ViT-B-32 (Radford et al., 2021) Euclidean DataCompDr 12,779,520 1,000 1,536 Yes Text-to-Image coca ViT-L-14 (Yu et al., 2022) Euclidean AmazonBooks 15,928,208 1,000 384 No Text all-MiniLM-L12-v2 (Wang et al., 2020) Euclidean MajorTOM 56,506,400 10,000 1,024 No Image DINOv2 (Oquab et al., 2024) Euclidean Legacy Datasets Word2Vec 1,000,000 1,000 300 No Text Word2Vec (Mikolov et al., 2013) Euclidean GIST 1,000,000 1,000 960 No Image GIST (Oliva & Torralba, 2001) Euclidean GloVe 1,193,514 1,000 200 No Text GloVe (Pennington et al., 2014) Euclidean ImageNet 2,340,373 200 150 No Image dense SIFT (Lazebnik et al., 2006) Euclidean SIFT10M 10,000,000 1,000 128 No Image SIFT (Lowe, 2004) Cosine DEEP100M 100,000,000 1,000 96 No Image GoogLeNet (Szegedy et al., 2015) Cosine
+
+[p. 8 | section: 4.4. Parameter Analysis of PAG | type: Text]
+patible with the AVX512 instruction set. The choice of 4 bits follows NGT-QG (Yahoo! Japan, 2023) .
+
+[p. 8 | section: 4.4. Parameter Analysis of PAG | type: ListGroup]
+(2) L: L is recommended to be in [8, d/8], where 8 is compatible with AVX512 because at most 8 levels can be accessed at a time, while d/L = 8 is a safe ratio that ensures the accuracy of the multi-level projection structure, as explained in PEOs (Lu et al., 2024) . Based on the analysis in Sec. 4.3, L is used to tune the trade-off between efficiency and accuracy. (3) M: Similar to its usage in HNSW, 2M is the maximum out-degree of PAG. Based on the analysis in Sec. 4.3, M is recommended to be 64 when space cost permits. In practice, M is generally chosen from {16, 32, 64}. (4) |W|: The size of W is set to max{10, K}. Under this setting, when K ≥ 10, W is aligned with the result list, which enables refilling to be easily executed. (5) efC: efC determines how many nodes are visited in W and plays a role similar to efC in HNSW. It is used to control the trade-off between indexing time and search performance. For fast indexing, efC is recommended to be in [100, 200]. For high graph quality, efC is recommended to be in [1000, 10000]. Notably, thanks to TFB, PAG with a very large efC can be built much faster than HNSW with the same efC.
+
+[p. 8 | section: 5. Experiments | type: Text]
+We report main results here. Detailed setup and more results are available in Appendices D and E, respectively.
+
+[p. 8 | section: 5.1. Experimental Setup | type: Text]
+Data Statistics. In the main experiments, we evaluate PAG on eight datasets: DBpedia1536 (Qdrant, 2024a) (a.k.a., OpenAI-1536), DBpedia3072 (Qdrant, 2024b) (a.k.a., OpenAI-3072), WoltFood (Qdrant, 2024c) , AmazonBooks (Kang et al., 2024) , DataCompDr (Apple, 2025) ,
+
+[p. 8 | section: 5.1. Experimental Setup | type: Text]
+MajorTOM (Major TOM, 2024) , GloVe (Pennington et al., 2014) , and DEEP100M (Simhadri et al., 2021) . The first six are modern text, image, and multimodal (text-to-image) datasets generated by recent embedding models. The latter two are widely-used legacy datasets (downloaded from The Similarity Search Team, CUHK (2018) ). In addition, we report the experimental results on other legacy datasets (Word2Vec, GIST, ImageNet, and SIFT10M) in Appendix E.3. Dataset statistics are reported in Table 2. The queries of DataCompDr are out-of-distribution (OOD). Fig. 11 in Appendix D plots the data and query distributions of this dataset.
+
+[p. 8 | section: 5.1. Experimental Setup | type: Text]
+Baselines. Baselines include HNSW (Malkov & Yashunin, 2020) , Vamana (Subramanya et al., 2019) , SymQG (Gou et al., 2025) , ScaNN (Guo et al., 2020a) , IVFPQFS (Jegou ´ et al., 2011) , and RaBitQ+ (Gao et al., 2025) . We design two variants of PAG by tuning its parameters, as shown in Table 4 in Appendix D: (1) PAG-Base, for higher search performance, and (2) PAG-Lite, for faster indexing and smaller memory footprint.
+
+[p. 8 | section: 5.2. Experimental Results | type: Text]
+We set the default retrieval size K = 100.
+
+[p. 8 | section: 5.2. Experimental Results | type: ListGroup]
+D1. QPS-Recall. Fig. 3 plots the QPS-recall performance. PAG-Base performs the best on all the modern datasets except for high recall settings on WoltFood. Its speedup over HNSW can be up to 5 times. For legacy datasets, PAG-Base is the best on GloVe and only second to SymQG on DEEP100M. PAG-Lite also delivers competitive search performance and is the runner-up on DBpedia1536, DBpedia3072, and MajorTOM. D2. Indexing Time. Fig. 4 (left) shows the indexing time. Under the same or larger efC, PAG-Base requires only 20–40% of the indexing time of HNSW, and is faster than SymQG in most cases. PAG-Lite achieves an indexing time comparable to quantization-based methods, which is further reduced
+
+[p. 9 | section: 5.2. Experimental Results | type: FigureGroup]
+Figure 3. QPS-recall, K = 100. SymQG runs out of memory on MajorTOM. Recall is plotted in logarithmic scale to highlight large values.
+
+[p. 9 | section: 5.2. Experimental Results | type: FigureGroup]
+Figure 4. Indexing time and peak memory usage. SymQG runs out of memory on MajorTOM.
+
+[p. 9 | section: 5.2. Experimental Results | type: Text]
+to 0.5 \times on high-dimensional datasets, where it attains the lowest indexing time.
+
+[p. 9 | section: 5.2. Experimental Results | type: Text]
+D3. Memory Footprint. Fig. 4 shows the memory usage in indexing (middle) and searching (right) phases. PAG-Base uses more memory than HNSW on lower-dimensional datasets, but the difference is negligible on higher-dimensional datasets. PAG-Base consistently uses much less memory than SymQG. PAG-Lite achieves the smallest memory footprint in 4 out of 8 cases for both indexing and searching.
+
+[p. 9 | section: 5.2. Experimental Results | type: Text]
+D4. High-dimensional Scalability. The competitiveness of PAG-Base and PAG-Lite over d \in [96,3072] , as shown in Figs. 3 and 4, is consistent, showcasing its insensitivity to dimensionality. In addition, the advantage of PAG in QPS-recall becomes more pronounced on high-dimensional datasets. In contrast, SymQG reports very low recall on high-dimensional datasets DBpedia1536, DBpedia3072, and DataCompDr, where d \in \{1536,3072\} .
+
+[p. 9 | section: 5.2. Experimental Results | type: Text]
+\underline{D5.\ Retrieval\ Size\ Robustness} . Besides the QPS-recall for K=100 in Fig. 3, we report the results for K=10 in Fig. 16 and K=1000 in Fig. 18. PAG methods are highly
+
+[p. 9 | section: 5.2. Experimental Results | type: Text]
+competitive across the three K values. When K=10, PAG-Base achieves comparable performance and even outperforms SymQG in the high-recall ( \geq 95\% ) region on all datasets except DEEP100M. When K=1000, PAG-Base remains the best, while the performance of SymQG degrades significantly. To highlight the performance comparison, we also plot in Fig. 7 the QPS-recall by varying K from 200 to 800. It can be seen that the gap between PAG-Base and SymQG grows when K moves towards larger values. This observation showcases the robustness of PAG in various applications that differ in retrieval size.
+
+[p. 9 | section: 5.2. Experimental Results | type: Text]
+D6. Online Insertion Support. To evaluate the query processing performance with online insertions, we consider the following workload. We randomly sample from the corpus 10,000 vectors as insertion queries and another 10,000 vectors as search queries. The 20,000 vectors are divided into 20 batches, each with 1,000 vectors. The insertion batches and the search batches are interleaved as a workload, with insertion as the first batch. The rest of the corpus is used to build the initial index. Note that DataCompDr is not OOD in this setting because its original query set (in Table 2) is
+
+[p. 10 | section: 5.2. Experimental Results | type: FigureGroup]
+Figure 6. QPS-recall, K = 1000.
+
+[p. 10 | section: 5.2. Experimental Results | type: Text]
+not used. To make the processing times of insertion queries and search queries on the same scale, we set efS = efC and tune these two parameters to control the recall. Fig. 8 compares PAG-Base and HNSW, plotting their QPS-recall performances of insertion and search. Insertion queries are slightly slower to process than search queries for both methods. PAG-Base are much faster than HNSW in both insertion and search speeds, and PAG-Base's insertion is even faster than HNSW's search. Similar to PAG-Base's advantage in search, its speedup over HNSW in insertion can be up to 5 times, demonstrating its efficiency in processing online insertions.
+
+[p. 10 | section: 5.2. Experimental Results | type: Text]
+Ablation Study. We choose four datasets, DBpedia3072, WoltFood, AmazonBooks, and DataCompDr, for ablation study. Among the six modern datasets, they cover the lowest and highest dimensionality, text and images, as well as OOD. Figs. 9 and 10 show the effectiveness of PAG's components. We observe that TFB consistently reduces indexing time and improves search performance. PES further enhances search performance with negligible additional indexing time. The additional memory usage introduced by TFB and PES is very minor and thus not shown here, because TFB does not affect
+
+[p. 10 | section: 5.2. Experimental Results | type: Text]
+the index size and PES barely increases the number of edges.
+
+[p. 10 | section: 6. Conclusion | type: Text]
+In this paper, we motivated six critical demands of modern AI applications for ANNS, covering search performance, indexing speed, memory footprint, scalability to dimensionality, robustness against retrieval size, and support of online insertion. To meet these demands, we proposed PAG as a new framework for ANNS. PAG reduces unnecessary distance computations by employing the comparison of exact and approximate distances. The components of PAG are derived from a unified statistical relationship, making its mechanism theoretically explainable. Experiments on modern datasets showcased the superiority of PAG over widely-used ANNS methods as well as state-of-the art solutions, and confirmed the effectiveness of PAG's components.

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+# Approximate Nearest Neighbor Search for Modern AI: A Projection-Augmented Graph Approach
+
+## Anonymous Authors<sup>1</sup>
+
+## Abstract
+
+Approximate Nearest Neighbor Search (ANNS) is fundamental to modern AI applications. Most existing solutions optimize query efficiency but fail to align with the practical requirements of modern workloads. In this paper, we outline six critical demands of modern AI applications: high query efficiency, fast indexing, low memory footprint, scalability to high dimensionality, robustness across varying retrieval sizes, and support for online insertions. To satisfy all these demands, we introduce Projection-Augmented Graph (PAG), a new ANNS framework that integrates projection techniques into a graph index. PAG reduces unnecessary exact distance computations through asymmetric comparisons between exact and approximate distances as guided by projection-based statistical tests. Three key components are designed and unified to the graph index to optimize indexing and searching. Experiments on six modern datasets demonstrate that PAG consistently achieves superior query per second (QPS) recall performance—up to 5×faster than HNSW while offering fast indexing speed and moderate memory footprint. PAG remains robust as dimensionality and retrieval size increase and naturally supports online insertions. Our source code is available at: [https://github.com/](https://github.com/KejingLu-810/PAG/) [KejingLu-810/PAG/](https://github.com/KejingLu-810/PAG/) .
+
+## 1. Introduction
+
+Given a dataset D ⊂ R <sup>d</sup> with size n and a query q ∈ R d , Approximate Nearest Neighbor Search (ANNS) aims to find K approximate nearest neighbors of q as accurately and efficiently as possible. Due to its crucial role in many applications such as image search, recommender systems, and
+
+Preliminary work. Under review by the International Conference on Machine Learning (ICML). Do not distribute.
+
+retrieval-augmented-generation (RAG), we have witnessed the rapid proliferation of ANNS solvers. Whereas the ANN-Benchmarks [\(Bernhardsson et al.,](#page-11-0) [2015\)](#page-11-0) has been developed to compare these methods in a unified environment, many datasets in the ANN-Benchmarks, as pointed out by [Chen](#page-11-1) [et al.](#page-11-1) [\(2025\)](#page-11-1), are derived from outdated models such as SIFT and GIST, which are image descriptors developed more than 20 years ago [\(Lowe,](#page-12-0) [2004;](#page-12-0) [Oliva & Torralba,](#page-12-1) [2001\)](#page-12-1). In addition, its evaluation metric is limited to query per second (QPS)-recall under K = 10.
+
+Seeing the fast development of AI technology, we argue that a well-designed ANNS solver should perform well on modern datasets, and beyond (D1) QPS-recall performance, several demands must also be considered: (D2) indexing time, for which graph indexes such as HNSW [\(Malkov & Yashunin,](#page-12-2) [2020\)](#page-12-2) are often criticized [\(Lin,](#page-12-3) [2025;](#page-12-3) [Li & Papakonstantinou,](#page-12-4) [2025\)](#page-12-4), compromising their use in applications that require instant deployment; (D3) memory footprint, where moderate and adjustable consumption, mostly caused by the index size, is desired for the memory vs accuracy trade-off [\(Cheng](#page-11-2) [et al.,](#page-11-2) [2024;](#page-11-2) [Wallace,](#page-13-0) [2025\)](#page-13-0); (D4) scalability to high dimensionality, as motivated by the increasing dimensionality of modern embedding models such as CLIP [\(Radford et al.,](#page-12-5) [2021\)](#page-12-5) [1](#page-0-0) ; (D5) robustness against the retrieval size K, due to the needs in various applications (e.g., K is typically 10 for RAG [\(Fensore et al.,](#page-11-3) [2025;](#page-11-3) [Ke et al.,](#page-11-4) [2025\)](#page-11-4) but can be up to hundreds in image retrieval [\(Vendrow et al.,](#page-13-1) [2024\)](#page-13-1) and thousands in recommender systems [\(Zhang et al.,](#page-13-2) [2025\)](#page-13-2)); (D6) support of online insertions, which is essential for emergent applications such as self-evolving agents that continually accumulate and reuse experience through interaction [\(Ouyang](#page-12-6) [et al.,](#page-12-6) [2025;](#page-12-6) [Zhai et al.,](#page-13-3) [2025;](#page-13-3) [Zhang et al.,](#page-13-4) [2026\)](#page-13-4).
+
+## <span id="page-0-1"></span>1.1. Prior Works
+
+We review representative ANNS solvers based on D1 – D6. More discussions on literature can be found in Appendix [C.](#page-15-0)
+
+<sup>1</sup>Anonymous Institution, Anonymous City, Anonymous Region, Anonymous Country. Correspondence to: Anonymous Author <anon.email@domain.com>.
+
+<span id="page-0-0"></span><sup>1</sup>Despite the availability of dimensionality reduction techniques, [Weller et al.](#page-13-5) [\(2025\)](#page-13-5) proved that for embedding-based retrieval with dimensionality d, there exists a dataset size n (e.g., n ≈ 4M for d = 1024) beyond which it is theoretically impossible to represent all possible top-2 document combinations, showcasing the necessity for higher dimensionality on large datasets.
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+<span id="page-1-0"></span>Table 1. Summary of empirical performance comparison of notable ANNS solvers, HNSW (Malkov & Yashunin, 2020), Vamana (in-memory DiskANN (Subramanya et al., 2019)), IVFPQFS (fast scan IVFPQ (Jégou et al., 2011)), ScaNN (Guo et al., 2020a), RaBitQ+ (Gao et al., 2025), SymQG (Gou et al., 2025), HNSW+KS2 (Lu et al., 2025), and our solutions PAG-Base (for high QPS) and PAG-Lite (for fast indexing and small index size). Detailed results are available in Sec. 5.2. The evaluation of D1 – D3 is clustered into tiers, where Tier-4 is the best. D4 – D6 are evaluated in yes/no (√/X).
+
+|                                                  | Graph-Based |        | Quantization-Based   |                    |              | QG     | PG       | PA       | ıG       |
+|--------------------------------------------------|-------------|--------|----------------------|--------------------|--------------|--------|----------|----------|----------|
+| Criteria                                         | HNSW        | Vamana | IVFPQFS <sup>1</sup> | ScaNN <sup>1</sup> | RaBitQ+      | SymQG  | HNSW+KS2 | PAG-Base | PAG-Lite |
+| D1. QPS-recall                                   | Tier-2      | Tier-2 | Tier-1               | Tier-1             | Tier-1       | Tier-3 | Tier-3   | Tier-4   | Tier-3   |
+| <b>D2.</b> Indexing time                         | Tier-1      | Tier-1 | Tier-3               | Tier-3             | Tier-3       | Tier-2 | Tier-1   | Tier-2   | Tier-4   |
+| <b>D3.</b> Memory footprint <sup>2</sup>         | Tier-3      | Tier-3 | Tier-4               | Tier-4             | Tier-4       | Tier-1 | Tier-2   | Tier-3   | Tier-4   |
+| <b>D4.</b> High-dim. scalability                 | ✓           | ✓      | ✓                    | ✓                  | ✓            | X      | ✓        | ✓        | ✓        |
+| <b>D5.</b> Retrieval size robustness             | ✓           | ✓      | ✓                    | ✓                  | ✓            | Х      | ✓        | ✓        | ✓        |
+| <b>D6.</b> Online insertion support <sup>3</sup> | ✓           | ✓      | ✓                    | ✓                  | $\checkmark$ | X      | ×        | ✓        | ✓        |
+
+Reranking is enabled for IVFPOFS and ScaNN.
+
+Graph-Based Methods. These methods construct a similarity graph connecting nearby vectors and hop towards the neighborhood of q. Notable methods are HNSW (Malkov & Yashunin, 2020), NSG (Fu et al., 2019), and DiskANN (Subramanya et al., 2019). They are generally competitive in QPS-recall but slow in building indexes.
+
+Quantization-Based Methods. These methods compress vectors and rank them using approximate (i.e., quantized) distances to q in pursuit of efficiency. Representative methods are IVFPQ (Jégou et al., 2011) and ScaNN (Guo et al., 2020a). They are fast in building indexes and save memory, but their QPS-recall is generally inferior to graph-based methods.
+
+Projection-Based Methods. These methods use random (e.g., E2LSH (Andoni & Indyk, 2005), Falconn (Andoni et al., 2015), and CEOs (Pham, 2021)) or data-dependent (e.g., learning to hash (Wang et al., 2018)) projections of the original vectors for indexing. Although many of them enjoy theoretical guarantees, they are less competitive than graphand quantization-based methods in QPS-recall performance, hence becoming less popular in modern top-K ANNS.
+
+<u>Tree-Based Methods.</u> They utilize tree indexes. table methods include k-d tree (Bentley, 1990), cover tree (Beygelzimer et al., 2006), and Annoy (Bernhardsson, 2013). Like projection-based methods, they are less widely used for modern ANNS due to limited QPS-recall performance.
+
+As discussed above, graph-based methods are good at QPSrecall. But they compute exact distances between vectors, which is computationally costly. To address this weakness, recent advancements attempt to integrate other techniques, in particular, quantization or projection, into graphs.
+
+Quantized Graph (QG) Methods. They construct a similarity graph based on quantized vectors instead of the original ones, thereby replacing exact distances with approximate values. Representative methods are NGT-QG (Yahoo! Japan, 2023), LVQ (Aguerrebere et al., 2023), and SymphonyQG (Gou et al., 2025). QG methods can achieve very high QPS-recall performance, yet they are sensitive to the data distribution and many of them do not perform very well in D3 - D6.
+
+*Projection* + *Graph* (*PG*) *Methods*. Unlike QG, they operate on original vectors and reduce unnecessary exact distance computations. To this end, they employ projection techniques to test whether a neighbor needs to be explored (called routing tests). Representative methods are FINGER (Chen et al., 2023), PEOs (Lu et al., 2024), and KS2 (Lu et al., 2025). Despite QPS-recall improvement on top of graph-based methods, such improvement comes at the cost of indexing time, memory footprint, and support for online insertions.
+
+Table 1 compares notable methods implemented in leading vector databases and recent ones representing state of the art.
+
+#### 1.2. Our Solution
+
+We propose a new ANNS framework, Projection-Augmented Graph (PAG), which integrates projection into a similarity graph and achieves superior performance for all the six criteria. Unlike PG methods, PAG treats projection as a fundamental building block of graph construction rather than as a plug-in. The rationale of PAG is to accommodate both exact and approximate distances within a *unified* framework, addressing two key issues: (1) when distances shall be computed exactly and when they shall be approximated, and (2) how exact and approximate distances are compared during indexing and searching to enhance the performance.
+
+PAG significantly reduces the computational cost of searching (D1) and indexing (D2) by carefully determining whether exact distance computation is needed. Such procedure relies on asymmetric comparisons between exact and approximate distance values obtained from space-efficient and adjustable random-projection structures (D3), as opposed to symmetric comparisons in QG. Moreover, the asymmetric distance
+
+Memory footprint for searching, rather than indexing, is compared.
+
+<sup>&</sup>lt;sup>3</sup> For any streaming workload Q composed of search and insertion queries, we say an algorithm supports online insertion, if the amortized cost of processing a query in Q is O(1) times the cost of a search. In other words, the index can be incrementally updated with minimal cost. See Sec. 4.3 for the analysis of PAG.
+
+{2}------------------------------------------------
+
+comparisons, as guided by random-projection-based statistical tests, only need to tell which value is larger, thereby achieving accuracy while preserving efficiency. Meanwhile, PAG inherits graph-based methods' scalability to high dimensionality (D4) and robustness against retrieval size (D5), and follows the search-and-insertion paradigm (e.g., HNSW) to accommodate online insertions (D6).
+
+110 111
+
+114
+
+124
+
+126
+
+134
+
+136
+
+Our technical contributions are summarized as follows.
+
+- (1) Following the idea of routing tests in PEOs [\(Lu et al.,](#page-12-10) [2024\)](#page-12-10) and KS2 [\(Lu et al.,](#page-12-8) [2025\)](#page-12-8), we derive a Probabilistic Routing Test (PRT) function (Sec. [3.2\)](#page-4-0). The theoretical result (Theorem [3.1\)](#page-4-1), along with Lemma 4.3 in [Lu et al.](#page-12-8) [\(2025\)](#page-12-8), provides a complete theoretical explanation of PRT. In addition, our work is the first to apply PRT to graph construction.
+- (2) We propose a data structure called Test Feedback Buffer (TFB) (Sec. [3.3\)](#page-5-0), which refines the threshold setting in PRT and enables the reuse of false positives generated by PRT. By incorporating TFB into PRT, we obtain the PRT-TFB test as a core technique for accelerating both indexing and searching.
+- (3) We propose a statistical test called Probabilistic Edge Selection (PES) (Sec. [3.4\)](#page-5-1), which is derived from Theorem [3.1](#page-4-1) and can expand in-degrees when necessary. In collaboration with PRT, PRT-PES can improve the search performance for hard datasets on which traditional graph indexes perform poorly, while incurring very small indexing overhead.
+- (4) PRT, TFB, and PES constitute PAG (Fig. [1\)](#page-3-0). We show how the three components interact within PAG, with implementation details and complexity analysis provided.
+- (5) We conduct experiments on six modern (post-2023) datasets covering text, image, and multimodal data, with dimensionality ranging from 384 to 3072 and retrieval size from 10 to 1000. The results show that PAG achieves the best QPS-recall performance (up to 5× faster than HNSW), and its superiority is particularly evident on datasets with higher dimensionality. PAG remains dominant as K increases. By adjusting parameters, PAG can deliver the fastest indexing speed and lowest memory footprint on most datasets while maintaining competitive QPS-recall. PAG also performs well on the datasets in the ANN-Benchmarks, showcasing its compatibility with data generated by legacy models.
+
+## 2. Problem Setting
+
+Since PAG adopts a search-and-insertion paradigm, its search process, similar to HNSW, can be seen as a special case of index construction (i.e., no update to the graph), and we can focus on the problems that arise in index construction. Because constructing a similarity graph is essentially determining its edge set, we consider the following question: *Can we design statistical methods that select edges efficiently (i.e., fast indexing) and effectively (i.e., fast searching)?* To answer this question, we first review the existing edge-selection strategy, and then formalize the problems to be solved.
+
+#### 2.1. Background of RobustPrune
+
+Given a directed graph, let v be a node whose edges are to be determined. To identify its in-neighbor set Nin(v) and out-neighbor set Nout(v), state-of-the-art methods, such as HNSW, NSG and Vamana, employ a strategy known as RobustPrune to decide whether a candidate edge should be preserved. Take the construction of Nin(v) as an example. For a candidate neighbor u of node v, the pruning criterion considers the following set [2](#page-2-0) :
+
+$$S_{u,v} = \{ w \in N_{\text{out}}(u) \mid ||w - v|| \le ||v - u|| \}.$$
+ (1)
+
+In RobustPrune, if there exists w in Su,<sup>v</sup> such that ∥w − u∥ ≤ ∥v − u∥, u is not added to Nin(v); otherwise, Nin(v) is updated as Nin(v)∪{u}. This criterion admits an intuitive interpretation: if ∥w − u∥ ≤ ∥v − u∥ holds for some w, it is likely that one can reach v from u via w along a monotonic search path [\(Fu et al.,](#page-11-9) [2019\)](#page-11-9), implying that the direct edge uv⃗ is redundant. Similarly, Nout(v) can be constructed by RobustPrune with the roles of u and v reversed. In sum, existing graph-based methods determine Nout(v) and Nin(v) sequentially in the following manner.
+
+Nout(v)*.* We take v as the query and conduct an ANNS from v to obtain a candidate set, which is stored in a priority queue P. RobustPrune is then applied to P to determine Nout(v).
+
+Nin(v)*.* We check every u in Nout(v) and preserve those u's retained by RobustPrune as Nin(v).
+
+### 2.2. Towards Fast Graph Construction
+
+During graph construction, most of the time is spent on ANNS for the candidate set of Nout(v). Although we can reduce the graph construction parameter (e.g., efC in HNSW) for fast graph construction, the query processing performance degrades accordingly. Thus, we raise the following question.
+
+Q1. *How can we accelerate the ANNS for* v *while preserving the query processing performance?*
+
+Existing studies PEOs and KS2 have shown that PRT has the potential to be an appropriate answer to Q1. For each visited node u and its out-neighbor w, PRT sends (u, v, w) to a probabilistic test function to see whether they can pass the test, and then determines whether the exact distance between v and w needs to be computed based on the test result. Consequently, PRT avoids unnecessary distance computations, leading to faster search. In KS2 [\(Lu et al.,](#page-12-8) [2025\)](#page-12-8), it is shown that the routing test under ℓ<sup>2</sup> and cosine distances is equivalent to the angle-thresholding problem when vector norms
+
+<span id="page-2-0"></span>For Vamana, we assume its pruning ratio α = 1 and omit it in the inequality.
+
+{3}------------------------------------------------
+
+<span id="page-3-0"></span>![](_page_3_Figure_2.jpeg)
+
+Figure 1. An overview of PAG. In this example,  $\boldsymbol{u}$  has 7 out-neighbors  $\{\boldsymbol{w_i}\}_{i=1}^7$ . Let  $\{\boldsymbol{e_i}\}_{i=1}^7$  denote the edges between  $\boldsymbol{u}$  and  $\{\boldsymbol{w_i}\}_{i=1}^7$ , which are sent to PRT, where the threshold is determined by  $\boldsymbol{z_{max}}$ . As a result, only  $\boldsymbol{w_3}$  and  $\boldsymbol{w_5}$  pass PRT, and their exact distances to  $\boldsymbol{v}$  are computed. By distance comparison, node  $\boldsymbol{w_5}$  is identified as a false positive, not added to W, and thus sent to  $R_F$ . Node  $\boldsymbol{w_3}$  is inserted into W, causing  $\boldsymbol{z_{max}}$  to be ejected from W and sent to  $R_T$ . The two rings  $R_F$  and  $R_T$  are merged and refilled into W. On the other route, by a left shift, the signs of both  $\boldsymbol{w_3}$  and  $\boldsymbol{w_5}$  are reversed, and all signs become negative, indicating that  $\boldsymbol{u}$  is rejected by the PES test. Consequently,  $\boldsymbol{u}\dot{\boldsymbol{v}}$  is treated as a candidate edge and added to the PES set.
+
+are stored explicitly. Therefore, PRT can be formulated as follows:
+
+<span id="page-3-2"></span>**Problem 2.1.** (**Probabilistic Routing Test**) Given  $u, v, w \in \mathbb{S}^{d-1}$  and a threshold  $-1 \le \tau \le 1$ , construct a random-projection vector set  $\mathcal{F}$  and design a test function w.r.t  $\mathcal{F}$ , i.e., PRT :  $(u, v, w, \tau) \to RV$  with time complexity o(d), where RV denotes the set of one-dimensional random variables, such that if  $\cos(\angle(v-u, w-u)) \ge \tau$ , then  $\mathbb{P}[\operatorname{PRT}(u, v, w, \tau) \ge 0] \ge 0.5$ . Otherwise,  $\mathbb{P}[\operatorname{PRT}(u, v, w, \tau) < 0] \to 1$  as  $|\mathcal{F}| \to \infty$ .
+
+Different from the routing test functions used in PEOs and KS2, we take  $\tau$  as an explicit input. In this paper, we use another method to determine  $\tau$  in order to further improve the efficiency of the routing test.
+
+#### 2.3. Towards High Graph Connectivity
+
+From the process of determining  $N_{\rm in}(\boldsymbol{v})$ , we observe that its candidate set is restricted to  $N_{\rm out}(\boldsymbol{v})$ . Recent work (Wang et al., 2025a) shows that  $|N_{\rm out}(\boldsymbol{v})|$  may be too small to reliably determine incoming edges, causing some nodes to have very small in-degrees. Such nodes may become unreachable during search, which partly explains why HNSW performs poorly on certain real-world datasets. On the other hand, for each checked node  $\boldsymbol{u}$ , the worst-case time complexity of RobustPrune can reach  $O(|N_{\rm out}(\boldsymbol{u})|d)$ , implying that a straightforward enlargement of the candidate set for
+
+<span id="page-3-1"></span>![](_page_3_Figure_9.jpeg)
+
+Figure 2. Illustration of PES (left: the role of PES; right: geometric illustration). Let  $\boldsymbol{v}$  be the node to be inserted. The four green nodes are out-neighbors of  $\boldsymbol{v}$ . Without PES, we can obtain only a single in-neighbor via RobustPrune. By taking all other visited nodes (the blue ones) into account, we apply PES, followed by RobustPrune. As such, we can identify three additional promising in-neighbors, thereby strengthening the connectivity of the neighborhood of  $\boldsymbol{v}$ .
+
+determining incoming edges is expensive. Considering this dilemma, we raise the following question.
+
+Q2. How can we efficiently detect incoming edges outside  $N_{\text{out}}(v)$  that are useful for improving query processing performance but hard to be identified by RobustPrune?
+
+{4}------------------------------------------------
+
+To answer this question, we formulate a problem as follows.
+
+<span id="page-4-2"></span>**Problem 2.2.** (**Probabilistic Edge Selection**) Given  $u, v \in \mathbb{S}^{d-1}$ , construct a random-projection-vector set  $\mathcal{F}$  and design a probabilistic edge selection function PES:  $(u, v) \to RV$  whose time complexity is  $O(|N_{\mathrm{out}}(u)|)$ , such that if  $S_{u,v}$  is non-empty, then  $\mathbb{P}[\mathrm{PES}(u,v) \geq 0] \geq 0.5$ . Otherwise,  $\mathbb{P}[\mathrm{PES}(u,v) < 0] \to 1$  as  $|\mathcal{F}| \to \infty$ .
+
+We emphasize that the PES test is far more than a fast probabilistic implementation of RobustPrune. This is because the PES test can be applied to all visited nodes during the ANNS of  $\boldsymbol{v}$ , whose number is typically much larger than  $N_{\rm out}(\boldsymbol{v})$  in practice, leading to a better graph index in terms of connectivity (Fig. 2).
+
+## 3. Projection-Augmented Graph
+
+We show that Problems 2.1 and 2.2 can be solved in a unified framework PAG, which contains three components: Probabilistic Routing Test (PRT), Test Feedback Buffer (TFB), and Probabilistic Edge Selection (PES).
+
+### <span id="page-4-4"></span>3.1. Neighborhood Relations via Random Projection
+
+We present an asymptotic result that characterizes the relationship between multiple angles in high-dimensional spaces and their corresponding projection values onto a certain projection vector. This result forms the theoretical basis for all the three components. We first describe how to construct the random-projection vector set  $\mathcal F$  that appears in Problems 2.1 and 2.2, following an approach similar to that used in KS2 (Lu et al., 2025). Specifically, we divide the original space  $\mathbb R^d$  into L subspaces, each of dimension d/L. In each subspace, we generate multiple cross-polytopes and apply an independent rotation to each, producing a total of m normalized vectors on  $\mathbb S^{d/L-1}$ . By concatenation, we obtain a total of  $m^L$  normalized vectors  $\{r_j\}_{j=1}^{m^L}$ , which together form the set  $\mathcal F$ . Consider v as the node to be inserted and v as the candidate neighbor to be checked, let  $N_{\mathrm{out}}(u) := \{w_i\}_{i=1}^t$ . We define  $\{\alpha_i\}_{i=1}^t$  as follows.
+
+<span id="page-4-3"></span>
+$$\alpha_i := \arccos \frac{\langle \boldsymbol{w_i} - \boldsymbol{u}, \boldsymbol{v} - \boldsymbol{u} \rangle}{\|\boldsymbol{w_i} - \boldsymbol{u}\| \|\boldsymbol{v} - \boldsymbol{u}\|}, \quad 1 \le i \le t \quad (2)$$
+
+Since  $\|\boldsymbol{w_i} - \boldsymbol{u}\|$  and  $\|\boldsymbol{v} - \boldsymbol{u}\|$  can be pre-computed, we aim to estimate  $[\cos \alpha_1, \dots, \cos \alpha_t]^{\top}$  without exact inner product computation in Eq. (2). We will show that this can be realized by  $\mathcal{F}$ . For each  $\boldsymbol{w_i}$  ( $1 \leq i \leq t$ ), let  $\boldsymbol{r_i^*} \in \mathcal{F}$  be the reference vector that has the smallest angle with  $\boldsymbol{w_i} - \boldsymbol{u}$ , and denote this smallest angle by  $\beta_i$ . We use  $\cos \theta_i$  to denote the cosine of the angle between  $\boldsymbol{r_i^*}$  and  $\boldsymbol{v} - \boldsymbol{u}$ , and subscript l to denote the l-th sub-vector of the original vector in  $\mathbb{R}^d$ ,  $1 \leq l \leq L$ . We introduce the following assumptions for each i:
+
+(A1) 
+$$\|(\boldsymbol{w_i} - \boldsymbol{u})\| = 1$$
+ and  $\|(\boldsymbol{v} - \boldsymbol{u})\| = 1$ .
+
+(A2) 
+$$\|(\boldsymbol{w_i} - \boldsymbol{u})_l\|$$
+ and  $\|(\boldsymbol{v} - \boldsymbol{u})_l\|$  are equal to  $(1 + o(1))/\sqrt{L}$ .  
+(A3)  $\langle (\boldsymbol{v} - \boldsymbol{u})_l, (\boldsymbol{w_i} - \boldsymbol{u})_l \rangle = (\cos \alpha_i)(1 + o(1))/L$ .
+
+Here, A1 does not lose generality, and A2 and A3 are mild for large d and L (see the remarks in Appendix B). Under these assumptions, the following theorem shows that  $\{\alpha_i\}_{i=1}^t$  can be asymptotically estimated by  $\{\beta_i\}_{i=1}^t$  and  $\{\theta_i\}_{i=1}^t$ .
+
+<span id="page-4-1"></span>**Theorem 3.1.** Under A1–A3, as  $L \to \infty$ ,  $d/L \to \infty$ , and m grows sufficiently fast with respect to d/L,  $X = [\cos \theta_1, \dots, \cos \theta_t]^\top$ , conditioned on  $\{\alpha_i, \beta_i\}_{i=1}^t$ , is asymptotically Gaussian:
+
+$$X \mid \{\alpha_i, \beta_i\}_{i=1}^t \xrightarrow{d} \mathcal{N}(\bar{\mu}, \bar{\Sigma}_{m,L})$$
+ (3)
+
+where the mean is  $\bar{\mu} = [\cos \alpha_1 \cos \beta_1, \dots, \cos \alpha_t \cos \beta_t]^\top$ , and  $\bar{\Sigma}_{m,L} = O(\epsilon_m/L)$  with  $\epsilon_m \to 0$  as  $m \to \infty$ .
+
+**Remarks.** (1) Theorem 3.1 establishes a geometric relationship among multiple out-neighbors of  $\boldsymbol{u}$  and  $\boldsymbol{v}$ , allowing us to estimate the relative location of  $\boldsymbol{v}$  w.r.t.  $\boldsymbol{u}$  within the neighborhood of  $\boldsymbol{u}$ . This relationship is the key to both PRT and PES. (2) In our proof,  $m \to \infty$  is used to eliminate noise from off-diagonal entries, while  $L \to \infty$  is required for the central limit theorem (CLT). In practice, moderate values of m and L are sufficient. Weak results under finite (m, L) can be found in Lemma 4.3 in the KS2 paper (Lu et al., 2025). (3)  $\boldsymbol{X}$  can be computed efficiently by AVX512 on modern CPUs. Vector  $[\cos \beta_1, \ldots, \cos \beta_t]^{\top}$  can be pre-computed, meaning that we can obtain an efficient way to estimate all  $\cos \alpha_i$ 's simultaneously.
+
+#### <span id="page-4-0"></span>3.2. Probabilistic Routing Test
+
+Let  $\tau_i$  be a threshold w.r.t  $w_i$ . Our PRT function is as follows:
+
+<span id="page-4-5"></span>
+$$PRT(\boldsymbol{u}, \boldsymbol{v}, \boldsymbol{w_i}, \tau_i) = \frac{\cos \theta_i}{\cos \beta_i} - \tau_i.$$
+ (4)
+
+For fixed (u, v), if the value of PRT is positive, the corresponding  $w_i$  passes the PRT. Based on Theorem 3.1, under asymptotic assumptions, the PRT function satisfies all the required properties in Problem 2.1. Here, we do not specify the setting of  $\tau_i$ , which will be postponed to Sec. 3.3.
+
+**Remarks.** Following the use of routing test in PEOs and KS2, exact distances are computed only between v and those  $w_i$ 's that pass PRT (see Fig. 1). The PRT function has the same structure as the KS2 test function (Lu et al., 2025), except for the setting of the threshold  $\tau$ . However, since the PRT and KS2 tests are derived using different principles, their theoretical results are complementary. Theorem 4.3 in the KS2 paper shows that, for every single  $w_i$ , even for finite (m, L), if  $\cos(\angle(v - u, w_i - u)) \ge \tau_i$ , then  $\mathbb{P}[\operatorname{PRT}(u, v, w_i, \tau_i) \ge 0] \ge 0.5$  still holds. Our result, on the other hand, characterizes the concrete asymptotic distribution and reveals the impact of L on the covariance, thereby explaining how L is used to adjust the estimation accuracy.
+
+{5}------------------------------------------------
+
+## <span id="page-5-0"></span>3.3. Test Feedback Buffer
+
+275 276
+
+294
+
+296
+
+314
+
+316
+
+324
+
+326
+
+328 329 For the PRT function, a key issue is how to set an appropriate threshold τ . In PEOs and KS2, τ is simply determined by the current furthest point in the result list (priority queue) P. However, due to the existence of false positives (FPs) generated by PRT, some points may pass PRT but cannot be added to P, which implies that the exact distance computations w.r.t such points are redundant. On the other hand, the Gaussian distribution established in Theorem [3.1](#page-4-1) implies that, with high probability, the actual distances of these FPs to the query q are not much larger than the threshold τ . This observation naturally raises the following question: *can we incrementally increase the threshold* τ *so that the currently generated FPs can be reused in subsequent search process?*
+
+We give an affirmative answer to this question and propose TFB, which consists of four components: the top-K result list RL, the working set W, and two ring buffers R<sup>F</sup> and R<sup>T</sup> , where |W| = |R<sup>F</sup> | = |R<sup>T</sup> |. The search procedure is divided into multiple rounds. In the j-th round, we operate on W and adopt the standard Best-First Search (BFS) strategy to update its nodes. During the BFS over W, in addition to the nodes stored in W, there are two types of nodes whose exact distances to the query q are computed. The first type consists of nodes ejected from W; these nodes are inserted into R<sup>T</sup> . The second type consists of false-positive (FP) nodes that pass the PRT but cannot be added to W; these nodes are inserted into R<sup>F</sup> . After all nodes in W have been visited, we update R<sup>L</sup> and clear W. We then merge R<sup>F</sup> and R<sup>T</sup> and sort the combined list by distance to q. The sorted nodes are first inserted into W until it is full, and the remaining nodes are inserted into R<sup>T</sup> . Finally, R<sup>F</sup> is cleared. The ANNS is completed after several such rounds. In the j-th round, let zmax denote the furthest node in W. τ<sup>i</sup> in Eq. [\(4\)](#page-4-5) is set to be the threshold on the cosine of the angle between w<sup>i</sup> − u and v − u required for w<sup>i</sup> to be admitted into W, i.e.,
+
+<span id="page-5-2"></span>
+$$\tau_i = \frac{\|u - w_i\|^2 + \|u - v\|^2 - \|z_{\max} - v\|^2}{2\|v - u\|\|w_i - u\|}.$$
+ (5)
+
+*PRT-TFB Test.* We call the combination of Eq. [\(4\)](#page-4-5) and Eq. [\(5\)](#page-5-2) the PRT-TFB test. Compared with existing routing test methods such as PEOs and KS2, which operate on the whole priority queue P, the PRT-TFB test has the following two advantages. (1) For large efC or large efS—which denotes the result list size, same as the notation in HNSW—the size of W is much smaller than that of P, and the movement of elements in W is much faster than that in P. (2) Each FP has a good chance of being selected as the visited node in the future rounds only if it remains in either of two rings, ensuring that its exact distance to q can be utilized at a certain time point.
+
+#### <span id="page-5-1"></span>3.4. Probabilistic Edge Selection
+
+In RobustPrune, ∥w<sup>i</sup> − v∥ needs to be smaller than ∥v − u∥ to allow u to be retained. Let δ<sup>i</sup> := ∥w<sup>i</sup> − u∥/(2∥v − u∥) denote the threshold on the cosine of the angle between w<sup>i</sup> − u and v − u, such that ∥w<sup>i</sup> − v∥ < ∥v − u∥. Then, the PES function is designed as follows.
+
+<span id="page-5-3"></span>
+$$PES(\boldsymbol{u}, \boldsymbol{v}) = \max_{1 \le i \le t} \left( \frac{\cos \theta_i}{\cos \beta_i} - \delta_i \right).$$
+ (6)
+
+By Theorem [3.1,](#page-4-1) we can see that the PES function in Eq. [\(6\)](#page-5-3) satisfies the probabilistic properties in Problem [2.2.](#page-4-2) In practice, if the value in Eq. [\(6\)](#page-5-3) is negative, we say that (u, v) is rejected by PES, and this edge will be regarded as a promising candidate for further examination of RobustPrune.
+
+*PRT-PES Collaboration.* From their definitions, the difference between the two thresholds, that is, τ<sup>i</sup> − δ<sup>i</sup> , is a value independent of v − w and can be pre-computed. Thus, we can conduct the PES test based on the test results of the PRT, leading to an O(1) time complexity, which confirms the objective O(|Nout(u)|) stated in Problem [2.2.](#page-4-2)
+
+*PES Set.* For every u whose PES value is negative, we do not check uv⃗ by RobustPrune immediately. Instead, we add uv⃗ into a so-called PES set for later examination (see Fig. [1\)](#page-3-0).
+
+## 4. Implementation and Analysis
+
+#### 4.1. Implementation
+
+*Indexing Phase.* PAG inserts every v ∈ D sequentially into the graph. For each v, the process can be divided into the following three steps.
+
+Step 1. PAG performs ANNS equipped with the PRT-TFB test to obtain the elements stored in RL, and then uses RobustPrune to determine Nout(v) and Nin(v).
+
+Step 2. Let Cand(v) denote the set of all nodes visited during ANNS. PAG uses PRT-PES to select additional in-neighbors from Cand(v), thereby supplementing the PES set.
+
+Based on our preceding discussion, PRT-TFB ensures the efficiency of ANNS, while PRT-PES detects more promising edges. We finally select more edges from the PES set.
+
+Step 3. For each candidate edge in the PES set, we apply RobustPrune to determine if it can be added to the graph.
+
+*Searching Phase.* The query processing with PAG is exactly the PRT-TFB-based ANNS for a query q.
+
+*Online Insertion.* Like HNSW, PAG naturally supports online insertion. The PES set is checked only after enough new nodes have been inserted into the graph.
+
+{6}------------------------------------------------
+
+#### <span id="page-6-1"></span>330 331 334 336 338 339 340 341 342 343 344 345 346 347 348 349 350 351 354 356 358 359 360 361 362 363 364 365 Algorithm 1: Construction of PAG Input: G := ∅ is the graph to be built; D is the dataset; 2M is the maximum out-degree; L is the space partition size; efC is the maximum size of visited nodes; b is the size of W Output: G with node set D, edge set E, and inference structure I <sup>1</sup> PES set is an empty table to accommodate candidate edges; <sup>2</sup> The capacity of R<sup>L</sup> is set to efC; <sup>3</sup> Maximum round Rmax is set to efC/b; <sup>4</sup> Generate multi-level random-projection structure F = {rjl} (1 ≤ j ≤ m, 1 ≤ l ≤ L); <sup>5</sup> Select b initial nodes, connect them, and write the projection information w.r.t the connected edges to I. Let V denote the set of all initial nodes; <sup>6</sup> foreach v ∈ D\V do <sup>7</sup> Insert b nodes to W and compute their distances to v; <sup>8</sup> Compute all ⟨vl, rjl⟩'s and build a projection table; <sup>9</sup> for r = 1 to Rmax do <sup>10</sup> foreach unvisited u ∈ W do <sup>11</sup> foreach w in Nout(u) do <sup>12</sup> if w passes the PRT-TFB test then <sup>13</sup> Compute the distance between v and w; <sup>14</sup> if ∥w − v∥ < ∥zmax − v∥ then <sup>15</sup> Update W and insert zmax to R<sup>T</sup> ; <sup>16</sup> else <sup>17</sup> Insert w to R<sup>F</sup> ; <sup>18</sup> if (u, v) is rejected by the PRT-PES test then <sup>19</sup> Add (u, v) to the PES set; <sup>20</sup> Update the nodes in W to RL, and empty W; <sup>21</sup> Sort and merge R<sup>F</sup> and R<sup>T</sup> , and empty R<sup>F</sup> and R<sup>T</sup> ; <sup>22</sup> Refill W until it is full using the sorted elements, and refill R<sup>T</sup> with the remaining sorted nodes; <sup>23</sup> Apply RobustPrune to R<sup>L</sup> to compute Nout(v) and Nin(v);
+
+#### 4.2. Algorithm Pseudo-Codes of PAG
+
+The pseudo-codes of the PAG algorithms are presented in Algs. [1](#page-6-1) and [2,](#page-6-2) which represent the indexing and searching, respectively. Note that indexing is essentially the sequential insertion of all nodes. Similar to [Lu et al.](#page-12-10) [\(2024\)](#page-12-10), we store the computed projections in a projection table for lookup (Line 8, Alg. [1](#page-6-1) and Line 4, Alg. [2\)](#page-6-2).
+
+<sup>24</sup> Add the projection information of new edges to I; <sup>25</sup> Apply RobustPrune to the PES set, supplement Nin(v)'s, and add the projection information of new edges to I;
+
+## <span id="page-6-0"></span>4.3. Complexity Analysis
+
+374
+
+376
+
+*Time complexity.* Searching with HNSW is in O(n ′d)-time, where n ′ is the number of visited nodes during the search. Although a rigorous analysis of n ′ remains an open problem, n ′ can be roughly estimated as O(M log n), where 2M is the
+
+#### Algorithm 2: Search with PAG
+
+```
+Input: G(D, E, I) is the constructed similarity graph; F
+         is the random-projection structure; efS is the
+         result list size; q is the query; K is the retrieval size
+  Output: Top-K ANN nodes
+1 The capacity of RL is set to K;
+2 b = max{10, K};
+3 Maximum round Rmax is set to efS/b;
+4 Compute all ⟨ql, rjl⟩'s and build a projection table;
+5 Insert b elements to W and compute their distance to q;
+6 for r = 1 to Rmax do
+7 foreach unvisited u ∈ W do
+8 foreach w in Nout(u) do
+ 9 if w passes PRT-TFB test then
+10 Compute the distance between q and w;
+11 if ∥w − q∥ < ∥zmax − q∥ then
+12 Update W and insert zmax to RT ;
+13 else
+14 Insert w to RF ;
+15 Update the nodes in W to RL, and empty W;
+16 Sort and merge RF and RT , and empty RF and RT ;
+17 Refill W until it is full using the sorted elements, and
+       refill RT with the remaining sorted nodes;
+18 Return the top-K nodes in RL;
+```
+
+maximum out-degree. In contrast, the complexity of PAG is O(n ′L+γn′d), whereL ≪ d is the user-specified parameter and γ ≤ 0.5 denotes the ratio of nodes that passed the PRT-PES test relative to the total number of candidates checked. γ is small in practice (generally < 0.2), and is smaller than the ratio in PEOs and KS2, because TFB ensures that the threshold increases incrementally rather than being set to the furthest distance w.r.t. a full priority queue.
+
+As for indexing, due to the search-and-insertion paradigm, we focus on the insertion of v, for which the complexity is O(n ′L + γn′d + M2d + M dm). Here, O(M2d) is the complexity of node connection, and O(M dm) is used to compute projection information, both of which are much smaller than the ANNS part due to their independence of n ′ .
+
+*Space complexity.* The space complexity of searching with PAG is O(nd + nM + nML), where O(nd) corresponds to the original dataset, O(nM) corresponds to the edge set and O(nML) corresponds to the inference structure. In the indexing phase, since we adopt in-place search for insertion, we only require an additional space of at most O(Mn) to store the edges in PES set. To further reduce the memory footprint, we can follow LVQ [\(Aguerrebere et al.,](#page-9-0) [2023\)](#page-9-0) and represent floats with 2 bytes.
+
+### 4.4. Parameter Analysis of PAG
+
+We analyze the settings of the parameters in PAG.
+
+(1) m: The value of m is fixed to 16, so that each projection vector in a subspace is represented by 4 bits, which is com
+
+{7}------------------------------------------------
+
+*Table 2.* Dataset statistics.
+
+<span id="page-7-1"></span>Name/Source Dataset Size Query Size Dim. OOD Type Embedding Model Distance Measure Modern Datasets DBpedia1536 999,000 1,000 1,536 No Text text-embedding-3-large [\(OpenAI,](#page-12-11) [2024\)](#page-12-11) Euclidean DBpedia3072 999,000 1,000 3,072 No Text text-embedding-3-large [\(OpenAI,](#page-12-11) [2024\)](#page-12-11) Euclidean WoltFood 1,719,611 1,000 512 No Image clip-ViT-B-32 [\(Radford et al.,](#page-12-5) [2021\)](#page-12-5) Euclidean DataCompDr 12,779,520 1,000 1,536 Yes Text-to-Image coca ViT-L-14 [\(Yu et al.,](#page-13-9) [2022\)](#page-13-9) Euclidean AmazonBooks 15,928,208 1,000 384 No Text all-MiniLM-L12-v2 [\(Wang et al.,](#page-13-10) [2020\)](#page-13-10) Euclidean MajorTOM 56,506,400 10,000 1,024 No Image DINOv2 [\(Oquab et al.,](#page-12-12) [2024\)](#page-12-12) Euclidean Legacy Datasets Word2Vec 1,000,000 1,000 300 No Text Word2Vec [\(Mikolov et al.,](#page-12-13) [2013\)](#page-12-13) Euclidean GIST 1,000,000 1,000 960 No Image GIST [\(Oliva & Torralba,](#page-12-1) [2001\)](#page-12-1) Euclidean GloVe 1,193,514 1,000 200 No Text GloVe [\(Pennington et al.,](#page-12-14) [2014\)](#page-12-14) Euclidean ImageNet 2,340,373 200 150 No Image dense SIFT [\(Lazebnik et al.,](#page-11-13) [2006\)](#page-11-13) Euclidean SIFT10M 10,000,000 1,000 128 No Image SIFT [\(Lowe,](#page-12-0) [2004\)](#page-12-0) Cosine DEEP100M 100,000,000 1,000 96 No Image GoogLeNet [\(Szegedy et al.,](#page-12-15) [2015\)](#page-12-15) Cosine
+
+patible with the AVX512 instruction set. The choice of 4 bits follows NGT-QG [\(Yahoo! Japan,](#page-13-7) [2023\)](#page-13-7).
+
+- (2) L: L is recommended to be in [8, d/8], where 8 is compatible with AVX512 because at most 8 levels can be accessed at a time, while d/L = 8 is a safe ratio that ensures the accuracy of the multi-level projection structure, as explained in PEOs [\(Lu et al.,](#page-12-10) [2024\)](#page-12-10). Based on the analysis in Sec. [4.3,](#page-6-0) L is used to tune the trade-off between efficiency and accuracy.
+- (3) M: Similar to its usage in HNSW, 2M is the maximum out-degree of PAG. Based on the analysis in Sec. [4.3,](#page-6-0) M is recommended to be 64 when space cost permits. In practice, M is generally chosen from {16, 32, 64}.
+- (4) |W|: The size of W is set to max{10, K}. Under this setting, when K ≥ 10, W is aligned with the result list, which enables refilling to be easily executed.
+- (5) efC: efC determines how many nodes are visited in W and plays a role similar to efC in HNSW. It is used to control the trade-off between indexing time and search performance. For fast indexing, efC is recommended to be in [100, 200]. For high graph quality, efC is recommended to be in [1000, 10000]. Notably, thanks to TFB, PAG with a very large efC can be built much faster than HNSW with the same efC.
+
+## 5. Experiments
+
+We report main results here. Detailed setup and more results are available in Appendices [D](#page-17-0) and [E,](#page-19-0) respectively.
+
+#### 5.1. Experimental Setup
+
+*Data Statistics.* In the main experiments, we evaluate PAG on eight datasets: DBpedia1536 [\(Qdrant,](#page-12-16) [2024a\)](#page-12-16) (a.k.a., OpenAI-1536), DBpedia3072 [\(Qdrant,](#page-12-17) [2024b\)](#page-12-17) (a.k.a., OpenAI-3072), WoltFood [\(Qdrant,](#page-12-18) [2024c\)](#page-12-18), AmazonBooks [\(Kang et al.,](#page-11-14) [2024\)](#page-11-14), DataCompDr [\(Apple,](#page-10-3) [2025\)](#page-10-3),
+
+MajorTOM [\(Major TOM,](#page-12-19) [2024\)](#page-12-19), GloVe [\(Pennington et al.,](#page-12-14) [2014\)](#page-12-14), and DEEP100M [\(Simhadri et al.,](#page-12-20) [2021\)](#page-12-20). The first six are modern text, image, and multimodal (text-to-image) datasets generated by recent embedding models. The latter two are widely-used legacy datasets (downloaded from [The Similarity Search Team, CUHK](#page-12-21) [\(2018\)](#page-12-21)). In addition, we report the experimental results on other legacy datasets (Word2Vec, GIST, ImageNet, and SIFT10M) in Appendix [E.3.](#page-19-1) Dataset statistics are reported in Table [2.](#page-7-1) The queries of DataCompDr are out-of-distribution (OOD). Fig. [11](#page-17-1) in Appendix [D](#page-17-0) plots the data and query distributions of this dataset.
+
+*Baselines.* Baselines include HNSW [\(Malkov & Yashunin,](#page-12-2) [2020\)](#page-12-2), Vamana [\(Subramanya et al.,](#page-12-7) [2019\)](#page-12-7), SymQG [\(Gou](#page-11-8) [et al.,](#page-11-8) [2025\)](#page-11-8), ScaNN [\(Guo et al.,](#page-11-6) [2020a\)](#page-11-6), IVFPQFS [\(Jegou](#page-11-5) ´ [et al.,](#page-11-5) [2011\)](#page-11-5), and RaBitQ+ [\(Gao et al.,](#page-11-7) [2025\)](#page-11-7). We design two variants of PAG by tuning its parameters, as shown in Table [4](#page-17-2) in Appendix [D:](#page-17-0) (1) PAG-Base, for higher search performance, and (2) PAG-Lite, for faster indexing and smaller memory footprint.
+
+#### <span id="page-7-0"></span>5.2. Experimental Results
+
+We set the default retrieval size K = 100.
+
+- *D1. QPS-Recall.* Fig. [3](#page-8-0) plots the QPS-recall performance. PAG-Base performs the best on all the modern datasets except for high recall settings on WoltFood. Its speedup over HNSW can be up to 5 times. For legacy datasets, PAG-Base is the best on GloVe and only second to SymQG on DEEP100M. PAG-Lite also delivers competitive search performance and is the runner-up on DBpedia1536, DBpedia3072, and MajorTOM.
+- *D2. Indexing Time.* Fig. [4](#page-8-1) (left) shows the indexing time. Under the same or larger efC, PAG-Base requires only 20–40% of the indexing time of HNSW, and is faster than SymQG in most cases. PAG-Lite achieves an indexing time comparable to quantization-based methods, which is further reduced
+
+{8}------------------------------------------------
+
+<span id="page-8-0"></span>![](_page_8_Figure_1.jpeg)
+
+<span id="page-8-1"></span>Figure 3. QPS-recall, K = 100. SymQG runs out of memory on MajorTOM. Recall is plotted in logarithmic scale to highlight large values.
+
+![](_page_8_Figure_3.jpeg)
+
+Figure 4. Indexing time and peak memory usage. SymQG runs out of memory on MajorTOM.
+
+to  $0.5 \times$  on high-dimensional datasets, where it attains the lowest indexing time.
+
+D3. Memory Footprint. Fig. 4 shows the memory usage in indexing (middle) and searching (right) phases. PAG-Base uses more memory than HNSW on lower-dimensional datasets, but the difference is negligible on higher-dimensional datasets. PAG-Base consistently uses much less memory than SymQG. PAG-Lite achieves the smallest memory footprint in 4 out of 8 cases for both indexing and searching.
+
+D4. High-dimensional Scalability. The competitiveness of PAG-Base and PAG-Lite over  $d \in [96,3072]$ , as shown in Figs. 3 and 4, is consistent, showcasing its insensitivity to dimensionality. In addition, the advantage of PAG in QPS-recall becomes more pronounced on high-dimensional datasets. In contrast, SymQG reports very low recall on high-dimensional datasets DBpedia1536, DBpedia3072, and DataCompDr, where  $d \in \{1536,3072\}$ .
+
+ $\underline{D5.\ Retrieval\ Size\ Robustness}$ . Besides the QPS-recall for K=100 in Fig. 3, we report the results for K=10 in Fig. 16 and K=1000 in Fig. 18. PAG methods are highly
+
+competitive across the three K values. When K=10, PAG-Base achieves comparable performance and even outperforms SymQG in the high-recall ( $\geq 95\%$ ) region on all datasets except DEEP100M. When K=1000, PAG-Base remains the best, while the performance of SymQG degrades significantly. To highlight the performance comparison, we also plot in Fig. 7 the QPS-recall by varying K from 200 to 800. It can be seen that the gap between PAG-Base and SymQG grows when K moves towards larger values. This observation showcases the robustness of PAG in various applications that differ in retrieval size.
+
+D6. Online Insertion Support. To evaluate the query processing performance with online insertions, we consider the following workload. We randomly sample from the corpus 10,000 vectors as insertion queries and another 10,000 vectors as search queries. The 20,000 vectors are divided into 20 batches, each with 1,000 vectors. The insertion batches and the search batches are interleaved as a workload, with insertion as the first batch. The rest of the corpus is used to build the initial index. Note that DataCompDr is not OOD in this setting because its original query set (in Table 2) is
+
+{9}------------------------------------------------
+
+![](_page_9_Figure_1.jpeg)
+
+Figure 6. QPS-recall, K = 1000.
+
+not used. To make the processing times of insertion queries and search queries on the same scale, we set efS = efC and tune these two parameters to control the recall. Fig. 8 compares PAG-Base and HNSW, plotting their QPS-recall performances of insertion and search. Insertion queries are slightly slower to process than search queries for both methods. PAG-Base are much faster than HNSW in both insertion and search speeds, and PAG-Base's insertion is even faster than HNSW's search. Similar to PAG-Base's advantage in search, its speedup over HNSW in insertion can be up to 5 times, demonstrating its efficiency in processing online insertions.
+
+Ablation Study. We choose four datasets, DBpedia3072, WoltFood, AmazonBooks, and DataCompDr, for ablation study. Among the six modern datasets, they cover the lowest and highest dimensionality, text and images, as well as OOD. Figs. 9 and 10 show the effectiveness of PAG's components. We observe that TFB consistently reduces indexing time and improves search performance. PES further enhances search performance with negligible additional indexing time. The additional memory usage introduced by TFB and PES is very minor and thus not shown here, because TFB does not affect
+
+the index size and PES barely increases the number of edges.
+
+## 6. Conclusion
+
+In this paper, we motivated six critical demands of modern AI applications for ANNS, covering search performance, indexing speed, memory footprint, scalability to dimensionality, robustness against retrieval size, and support of online insertion. To meet these demands, we proposed PAG as a new framework for ANNS. PAG reduces unnecessary distance computations by employing the comparison of exact and approximate distances. The components of PAG are derived from a unified statistical relationship, making its mechanism theoretically explainable. Experiments on modern datasets showcased the superiority of PAG over widely-used ANNS methods as well as state-of-the art solutions, and confirmed the effectiveness of PAG's components.
+
+#### References
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+
+{14}------------------------------------------------
+
+## A. Frequently Used Notations
+
+<span id="page-14-1"></span>Table 3 shows the notations frequently used in this paper. *Table 3.* Frequently used notations.
+
+| Symbol                             | Description                                                       |  |  |  |  |
+|------------------------------------|-------------------------------------------------------------------|--|--|--|--|
+| $\mathcal{D}$                      | Dataset of vectors                                                |  |  |  |  |
+| d                                  | Dimension of vectors                                              |  |  |  |  |
+| n                                  | Dataset size                                                      |  |  |  |  |
+| $\boldsymbol{q}$                   | Query vector                                                      |  |  |  |  |
+| K                                  | Retrieval size                                                    |  |  |  |  |
+| $\boldsymbol{v}$                   | Node to be inserted to a graph index                              |  |  |  |  |
+| $N_{\rm out}(\boldsymbol{v})$      | Out-neighbors of $v$                                              |  |  |  |  |
+| $N_{\rm in}(\boldsymbol{v})$       | In-neighbors of $\boldsymbol{v}$                                  |  |  |  |  |
+| $\boldsymbol{u}$                   | Candidate neighbor of $v$ to be evaluated                         |  |  |  |  |
+| L                                  | Space partition size                                              |  |  |  |  |
+| $\{\boldsymbol{w_i}\}_{i=1}^t$     | $t$ neighbors of $\boldsymbol{u}$                                 |  |  |  |  |
+| $\{\boldsymbol{e_i}\}_{i=1}^t$     | Edges between $\boldsymbol{u}$ and $\{\boldsymbol{w_i}\}_{i=1}^t$ |  |  |  |  |
+| $\tau_i$                           | Threshold w.r.t. $w_i$ in PRT                                     |  |  |  |  |
+| $\delta_i$                         | Threshold w.r.t. $w_i$ in PES                                     |  |  |  |  |
+| $R_L$                              | Result list                                                       |  |  |  |  |
+| W                                  | Working set                                                       |  |  |  |  |
+| $R_F, R_T$                         | Dual rings of false positives and ejected nodes                   |  |  |  |  |
+| m                                  | Number of projection vectors in each subspace                     |  |  |  |  |
+| ${\mathcal F}$                     | Set of concatenated projection vectors                            |  |  |  |  |
+| $\{\boldsymbol{r_i}\}_{i=1}^{m^L}$ | Individual projection vectors                                     |  |  |  |  |
+| $r_i^* \in \mathcal{F}$            | Vector having the smallest angle with $w_i - u$                   |  |  |  |  |
+| $\cos \alpha_i$                    | Cosine of the angle between $w_i - u$ and $v - u$                 |  |  |  |  |
+| $\cos \beta_i$                     | Cosine of the angle between $w_i - u$ and $r_i^*$                 |  |  |  |  |
+| $\cos \theta_i$                    | Cosine of the angle between $v-u$ and $r_i^*$                     |  |  |  |  |
+| Cand(v)                            | All nodes visited during the insertion of $v$                     |  |  |  |  |
+| $H(\boldsymbol{v})$                | Nodes in $\operatorname{Cand}(\boldsymbol{v})$ rejected by PES    |  |  |  |  |
+
+#### <span id="page-14-0"></span>B. Proof of Theorem 3.1
+
+In the proof, with slight abuse of notation, we treat u as the origin, and directly use v and each  $w_i$  to denote v-u and  $w_i-u$ , respectively.
+
+As such, we have  $w_1, \ldots, w_t, v \in \mathbb{S}^{d-1}$ , and  $\langle w_i, v \rangle = \cos \alpha_i, i = 1, \ldots, t$ . Suppose that d is divisible by L. Let  $w_i = [w_{i1}, \ldots, w_{iL}]^{\top}$  and  $v = [v_1, \ldots, v_L]^{\top}$  be the equal-dimension partition of  $w_i$  and v, respectively. Suppose that in the l-th subspace, we generate m random projection vectors  $\{r_{jl}\}_{j=1}^m$  with the norm  $1/\sqrt{L}$ . We use  $r_{il}^*$  to denote the nearest projection vector to  $w_{il}$ , i.e.,
+
+$$r_{il}^* = \underset{1 \le j \le m}{\arg\max} \langle r_{jl}, w_{il} \rangle. \tag{7}$$
+
+Then, we have  $\bm{r_i^*} = [\bm{r_{i1}^*}, \dots, \bm{r_{iL}^*}]^\top \in \mathbb{S}^{d-1}.$  We introduce
+
+$$C_{il} := \frac{\langle \boldsymbol{w_{il}}, \boldsymbol{r_{il}^*} \rangle \sqrt{L}}{\|\boldsymbol{w_{il}}\|} \in [-1, 1]. \tag{8}$$
+
+Let  $\mathbb{E}[C_{il}] = \mu_m$  and  $\mathrm{Var}(C_{il}) = \sigma_m^2$ , where  $\mu_m$  and  $\sigma_m^2$  depend only on m. Note that they implicitly depend on the subspace dimension d/L. When m grows at a sufficiently high rate compared to d/L,  $\mu_m \to 1$  and  $\sigma_m \to 0$ . Moreover,
+
+we have
+
+$$Cov(C_{il}, C_{jl}) = \rho_m(\phi_{ijl})\sigma_m^2 \tag{9}$$
+
+where  $\phi_{ijl}$  denotes the angle between  $w_{il}$  and  $w_{jl}$ , and  $\rho_m := [0,\pi] \to [-1,1]$  denotes the correlation coefficient function depending on  $\phi_{ijl}$ . Then, we have
+
+$$Y_i := \Sigma_{l=1}^L \langle \boldsymbol{r_{il}^*}, \boldsymbol{w_{il}} \rangle = \Sigma_{l=1}^L \frac{\|\boldsymbol{w_{il}}\|}{\sqrt{L}} C_{il}.$$
+ (10)
+
+By the assumption that  $\|\boldsymbol{w_{il}}\| = (1 + o(1))/\sqrt{L}$ , we have
+
+$$\mathbb{E}[Y_i] = \mu_m(1 + o(1)). \tag{11}$$
+
+$$Var(Y_i) = \frac{\sigma_m^2}{L}(1 + o(1)).$$
+ (12)
+
+<span id="page-14-2"></span>
+$$Cov(Y_i, Y_j) = \frac{\sigma_m^2}{L^2} \sum_{l=1}^{L} \rho_m(\phi_{ijl}) (1 + o(1)).$$
+ (13)
+
+Then,  $v_l$  can be decomposed as  $v_l = v_{il}^{\parallel} + v_{il}^{\perp}$ , where  $v_{il}^{\parallel}$  and  $v_{il}^{\perp}$  are defined as follows.
+
+$$v_{il}^{\parallel} = \frac{\langle v_l, w_{il} \rangle}{\|w_{il}\|^2} w_{il}, \quad v_{il}^{\perp} \perp w_{il}.$$
+ (14)
+
+Then, we can define  $Z_i$  as follows.
+
+$$Z_i := \langle r_i^*, v \rangle = Z_i^{(1)} + Z_i^{(2)}$$
+ (15)
+
+where  $Z_i^{(1)}$  and  $Z_i^{(2)}$  are defined as follows.
+
+$$Z_i^{(1)} := \sum_{l=1}^{L} \frac{\langle \boldsymbol{v_l}, \boldsymbol{w_{il}} \rangle}{\|\boldsymbol{w_{il}}\| \sqrt{L}} C_{il}.$$
+ (16)
+
+$$Z_i^{(2)} := \sum_{l=1}^L \langle r_{il}^*, v_{il}^{\perp} \rangle. \tag{17}$$
+
+$$\mathbb{E}[Z_i^{(1)}] = \frac{\mu_m}{\sqrt{L}} \sum_{l=1}^L \frac{\langle \boldsymbol{v}_l, \boldsymbol{w}_{il} \rangle}{\|\boldsymbol{w}_{il}\|}$$
+
+$$= \mu_m \cos \alpha_i + o(1).$$
+(18)
+
+By symmetry, we have  $\mathbb{E}[Z_i^{(2)}|\{C_{il}\}_{l=1}^L]=0$ . Then, by the independence of subspaces, we have
+
+$$\mathbb{E}[Z_i] = \mu_m \cos(\alpha_i) + o(1). \tag{19}$$
+
+$$Var(Z_i^{(1)}) = \frac{\sigma_m^2}{L} \sum_{l=1}^{L} \frac{\langle v_l, w_{il} \rangle^2}{\|w_{il}\|^2}.$$
+ (20)
+
+We take the orthogonal decomposition of  $r_{il}^*$  as  $r_{il}^* = r_{il}^{\parallel} + r_{il}^{\perp}$ , where  $r_{il}^{\parallel}$  has the same direction with  $w_{il}$ . Let  $r_{il}^{\perp} = \|r_{il}^{\perp}\|\zeta$ , where  $\|\zeta\| = 1$  is a random vector in a (d/L - 1)-dimensional subspace. We have
+
+$$\mathbb{E}_{\zeta}[\langle \zeta, v_{il}^{\perp} \rangle^{2}] = \frac{\|v_{il}^{\perp}\|^{2} L}{d - L}.$$
+ (21)
+
+{15}------------------------------------------------
+
+By symmetry,  $\mathbb{E}[\langle r_{il}^{\perp}, v_{il}^{\perp} \rangle \mid C_{il}] = 0$ . Then, we have
+
+$$\operatorname{Var}(\langle \boldsymbol{r}_{il}^*, \boldsymbol{v}_{il}^{\perp} \rangle) = \mathbb{E}_{C_{il}} \left[ \mathbb{E}[\langle \boldsymbol{r}_{il}^{\perp}, \boldsymbol{v}_{il}^{\perp} \rangle^2 \mid C_{il}] \right]. \tag{22}$$
+
+Using  $\mathbb{E}[\|\boldsymbol{r}_{il}^{\perp}\|^2] = \frac{1}{L}(1 - \mathbb{E}[C_{il}^2])$ , we have
+
+<span id="page-15-1"></span>
+$$Var(Z_i^{(2)}) = \frac{1 - (\sigma_m^2 + \mu_m^2)}{d - L} \sum_{l=1}^{L} \| v_{il}^{\perp} \|^2.$$
+ (23)
+
+Because  $Cov(Z_i^{(1)}, Z_i^{(2)}) = 0$ , we have
+
+$$Var(Z_i) = Var(Z_i^{(1)}) + Var(Z_i^{(2)}).$$
+ (24)
+
+We define  $\epsilon_m:=\sqrt{1-\mathbb{E}[C_{il}^2]}$ . From Eq. (23), we can see that  $\mathrm{Var}(Z_i^{(2)})=O(\epsilon_m^2/L)$ .
+
+We now analyze the covariance structure. We cannot assume  $Z^{(1)}$  and  $Z^{(2)}$  are uncorrelated across different indices  $i \neq j$ . Instead, we decompose the covariance matrix as
+
+<span id="page-15-3"></span>
+$$Cov(Z_i, Z_j) = Cov(Z_i^{(1)}, Z_j^{(1)}) + R_{ij}$$
+ (25)
+
+where the residual term  $R_{ij}$  contains the noise auto-covariance and cross-terms:
+
+$$R_{ij} = \text{Cov}(Z_i^{(2)}, Z_j^{(2)}) + \text{Cov}(Z_i^{(1)}, Z_j^{(2)}) + \text{Cov}(Z_i^{(2)}, Z_j^{(1)}).$$
+(26)
+
+Because  ${\rm Var}(Z^{(1)})=O(1/L)$  and  ${\rm Var}(Z^{(2)})=O(\epsilon_m^2/L)$ , by the Cauchy-Schwarz inequality, we have
+
+$$|\text{Cov}(Z_i^{(1)}, Z_j^{(2)})| \le \sqrt{\text{Var}(Z_i^{(1)})\text{Var}(Z_j^{(2)})} = O\left(\frac{\epsilon_m}{L}\right).$$
+
+Thus, the entire residual term satisfies  $R_{ij} = O(\epsilon_m/L)$ .
+
+Now, let us turn to  $\mathrm{Cov}(Y_i,Z_j)$ . When i=j, by conditional expectation,  $\mathrm{Cov}(Y_i,Z_j^{(2)})=0$ . When  $i\neq j$ , if  $m\to\infty$ ,  $\sigma_m^2+\mu_m^2\to 1$  and  $\mathrm{Var}(Z_i^{(2)})\to 0$ . Thus,  $\mathrm{Cov}(Y_i,Z_j^{(2)})\to 0$ . Then, as  $m\to\infty$ , we have
+
+<span id="page-15-2"></span>
+$$Cov(Y_i, Z_j) \to \frac{\sigma_m^2}{L} \sum_{l=1}^{L} \rho_m(\phi_{ijl}) \langle \boldsymbol{v_l}, \boldsymbol{w_{jl}} \rangle (1 + o(1)).$$
+(28)
+
+Based on the above analysis, we consider
+
+$$\boldsymbol{\xi_{l}} = \left[ \frac{\|\boldsymbol{w_{1l}}\|}{\sqrt{L}} C_{1l}, \dots, \frac{\|\boldsymbol{w_{tl}}\|}{\sqrt{L}} C_{tl}, \langle \boldsymbol{r_{1l}^*}, \boldsymbol{v_l} \rangle, \dots, \langle \boldsymbol{r_{tl}^*}, \boldsymbol{v_l} \rangle \right].$$
+(29)
+
+By the assumptions that  $\|v_{il}\|$  and  $\|w_{il}\|$  equal  $(1 + o(1))/\sqrt{L}$ , it can be seen that  $\xi_l$  satisfies the Lyapunov condition. By the independence of difference subspaces, we use the Lindeberg-Feller CLT and obtain the following result.
+
+$$\sqrt{L}([Y_1, \dots, Y_t, Z_1, \dots, Z_t]^{\top} - \bar{\boldsymbol{\mu}}) \xrightarrow[L \to \infty]{d} \mathcal{N}(0, \bar{\Sigma}_{m,L})$$
+(30)
+
+where  $\bar{\mu} = [\mu_m, \dots, \mu_m, \mu_m \cos \alpha_1, \dots, \mu_m \cos \alpha_t]^\top$ , and  $\bar{\Sigma}$  is represented by block sub-matrices
+
+$$\bar{\Sigma} = \begin{pmatrix} \Sigma_{YY} & \Sigma_{YZ} \\ \Sigma_{ZY} & \Sigma_{ZZ} \end{pmatrix} \tag{31}$$
+
+where the elements of  $\Sigma_{YY}$ ,  $\Sigma_{YZ}$  and  $\Sigma_{ZZ}$  are L times the covariance computed in Eqs. (13), (28), and (25), respectively.
+
+Based on this result, Z | Y is still a multivariate normal distribution. We have
+
+$$\mathbb{E}[\mathbf{Z}|\mathbf{Y}] = \boldsymbol{\mu}_Z + \boldsymbol{\Sigma}_{ZY} \boldsymbol{\Sigma}_{YY}^{-1} (\mathbf{Y} - \boldsymbol{\mu}_Y). \tag{32}$$
+
+$$Cov(\mathbf{Z}|\mathbf{Y}) = (\Sigma_{ZZ} - \Sigma_{ZY}\Sigma_{YY}^{-1}\Sigma_{YZ})/L.$$
+ (33)
+
+Let D be the diagonal matrix defined as follows
+
+$$\mathbf{D} = \operatorname{diag}[\cos \alpha_1, \dots, \cos \alpha_t]. \tag{34}$$
+
+Because  $\Sigma_{ZY} \to \mathbf{D}\Sigma_{YY}$ , when  $\mathbf{Y} = [\cos \beta_1, \dots, \cos \beta_t]^\top$ , we have
+
+$$\mathbb{E}[\mathbf{Z}|\mathbf{Y}] \to [\cos \alpha_1 \cos \beta_1, \dots, \cos \alpha_t \cos \beta_t]^\top.$$
+ (35)
+
+$$Cov(\mathbf{Z}|\mathbf{Y}) \to (\mathbf{\Sigma}_{\mathbf{Z}\mathbf{Z}} - \mathbf{D}\mathbf{\Sigma}_{\mathbf{Y}\mathbf{Y}}\mathbf{D})/L = \mathbf{R}/L$$
+ (36)
+
+where  $\mathbf{R}$  represents the scaled residual matrix with entries  $L \cdot R_{ij} = O(\epsilon_m)$ . Then, we conclude.
+
+**Remarks.** For each fixed i, if we apply a random rotation matrix to  $w_i - u$  and v - u in  $\mathbb{R}^d$ , by the spherical concentration inequality, we can derive the following inequalities.
+
+$$P\left(\left|\|(\boldsymbol{w_i} - \boldsymbol{u})_l\|^2 - \frac{1}{L}\right| \ge \tilde{\epsilon}\right) \le 2e^{-c_1 d\tilde{\epsilon}^2}$$
+ (37)
+
+$$P\left(\left|\langle (\boldsymbol{v}-\boldsymbol{u})_l, (\boldsymbol{w_i}-\boldsymbol{u})_l \rangle - \frac{\cos \alpha_i}{L}\right| \ge \tilde{\epsilon}\right) \le 2e^{-c_2 d\tilde{\epsilon}^2}$$
+(38)
+
+where  $c_1$  and  $c_2$  are constants. As  $L \to \infty$  and  $d/(L^2) \to \infty$ , we have the following results:
+
+$$\|(\boldsymbol{w_i} - \boldsymbol{u})_l\| = (1 + o_p(1))/\sqrt{L}.$$
+ (39)
+
+$$\langle (\boldsymbol{v} - \boldsymbol{u})_l, (\boldsymbol{w_i} - \boldsymbol{u})_l \rangle = \frac{\cos \alpha}{L} + o_p(\frac{1}{L}).$$
+ (40)
+
+This implies that  $L=\sqrt{d}$  is a balanced choice, which is consistent with the setting in our experiments.
+
+#### <span id="page-15-0"></span>C. Related Work
+
+We first supplement Sec. 1.1 with more discussions, and then introduce the preliminaries on probabilistic routing.
+
+{16}------------------------------------------------
+
+#### C.1. Discussions on ANNS Solvers
+
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+#### C.1.1. VECTOR QUANTIZATION
+
+Quantization-based methods for ANNS have a long history. Representative approaches can be found in (Jégou et al., 2011; Ge et al., 2014; Babenko & Lempitsky, 2016; Guo et al., 2020a; Gao & Long, 2024; Gao et al., 2025). Here, we briefly introduce the basic idea of the most widely-used method, Product Quantization (PQ) (Jégou et al., 2011).
+
+Let  $\mathcal{D} \subset \mathbb{R}^d$  be a d-dimensional dataset and let  $q \in \mathbb{R}^d$  be a query vector. PQ divides each data vector  $x \in \mathcal{D}$  into L sub-vectors, i.e.,  $x = (x_1, x_2, \ldots, x_L)$ , where each sub-vector  $x_j \in \mathbb{R}^{d'}$  has dimension d' = d/L. In this way, PQ constructs L subspaces of dimension d', each containing the corresponding sub-vectors of all data points.
+
+In each subspace, every sub-vector is assigned to one of  $2^k$  quantized sub-vectors. Equivalently, each sub-vector is represented by a sub-codeword of length k bits. The set of all sub-codewords in the j-th subspace is referred to as a codebook and is denoted by  $\mathcal{C}_j$ . By concatenating the sub-codewords across all subspaces, each original vector is represented by a codeword of length kL, taking values in the Cartesian product  $\mathcal{C}_1 \times \cdots \times \mathcal{C}_L$ .
+
+During the query phase, PQ first computes the distances between the query  $\boldsymbol{q}$  and all quantized sub-vectors in each subspace. For each data vector  $\boldsymbol{x}$ , PQ then sums the corresponding L subspace distances to obtain an approximate distance between  $\boldsymbol{x}$  and  $\boldsymbol{q}$ . After computing the approximate distances for all data vectors, PQ ranks them accordingly and returns the most promising candidates.
+
+### C.1.2. SIMILARITY GRAPH
+
+The workflow of graph-based methods is roughly as follows. In the indexing phase, a similarity graph is constructed as the index structure, where data points serve as nodes and edges connect pairs of nearby nodes. A search can move directly from one node to another only if an edge exists between them. In the query phase, the node with the highest priority in a priority queue is selected, and all of its connected neighbors are visited. During this process, the priority queue is updated whenever a node closer to the query is discovered. The search ends when all nodes in the priority queue have been visited. Finally, the top-K points in the priority queue are returned.
+
+Among various graph-based methods, HNSW (Malkov & Yashunin, 2020) is a notable one that employs a multi-layer hierarchical structure to achieve rapid routing. NSG (Fu et al., 2019) optimizes the graph topology to ensure the existence of monotonic paths towards a central entry point, thereby enhancing search efficiency. Vamana, which was introduced along with DiskANN (Subramanya et al., 2019), iteratively refines a random graph into a high-performance graph struc-
+
+ture. Despite their structural differences, these methods converge on a similar edge selection criterion, namely **RobustPrune**, which prioritizes directional diversity over simple proximity to ensure efficient navigation through high-dimensional space.
+
+In addition to the three graphs mentioned above, many graph-based methods have been proposed recently (Lu et al., 2021; Gao & Long, 2023; Xie et al., 2025; Wang et al., 2025b). Despite different designs, most of them rely on existing graphs, such as HNSW.
+
+#### C.1.3. QUANTIZED GRAPH
+
+As stated in the main body of this paper, QG can achieve high performance on some datasets for small values of K, and in certain cases can be  $4 \times -10 \times$  faster than HNSW. However, such improvement is often not robust and suffers from several limitations: (1) Sensitivity to data distribution. The effectiveness of QG relies on the assumption that the ranking induced by quantized distances is sufficiently close to the ranking under the original distances. Unfortunately, this assumption often does not hold, especially for many modern real-world datasets where semantic embeddings are increasingly complex. (2) **Sensitivity to** K. As K increases, the performance of QG degrades significantly. When K reaches the thousand scale, QG generally has no significant advantage over HNSW. (3) Very large space cost. To improve the accuracy of QG, more bits are required to represent the quantized vectors. As a result, the memory consumption of QG is typically at least  $2 \times$  larger than HNSW, compromising the use of QG for large datasets.
+
+#### C.2. Preliminaries of Routing Test
+
+#### C.2.1. RANDOM PROJECTION FOR ANGLE ESTIMATION
+
+Because the norms can be pre-computed, estimating the  $\ell_2$ distance is equivalent to estimating the cosine of the angle between vectors. In high-dimensional Euclidean spaces, angle estimation via random-projection techniques has been extensively studied, with Locality Sensitive Hashing (LSH) (Indyk & Motwani, 1998; Andoni & Indyk, 2005; 2008) being one of the most influential approaches. Among various LSH methods, SimHash (Charikar, 2002) is a representative one. Its core idea is to generate multiple random hyperplanes that partition the space into cells, so that vectors falling into the same cell are likely to form a small angle with each other. Subsequent studies have proposed more refined strategies for angular distance estimation. In particular, Andoni et al. (2015) introduced Falconn, an LSH method that identifies the projection vector yielding the largest or smallest projection value for a given data vector and uses the corresponding projection index as the hash value. This design leads to substantially improved search performance compared to SimHash. Building on this idea, Pham (2021) further incorporated Con-
+
+{17}------------------------------------------------
+
+<span id="page-17-1"></span>![](_page_17_Figure_1.jpeg)
+
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+Figure 11. DataCompDr data (sampled 50,000 vectors) and query (1,000 vectors) distributions.
+
+comitants of Extreme Order Statistics (CEOs) to explicitly identify the projection achieving the maximum or minimum inner product with the data vector, and to record the associated extreme projected value. By exploiting this additional information beyond a discrete hash value, a more accurate estimation of angular distance can be achieved (Pham & Liu, 2022).
+
+#### C.2.2. ROUTING TEST IN SIMILARITY GRAPHS
+
+Due to its simplicity and ease of implementation, CEOs has been adopted in a variety of similarity search tasks (Pham, 2021; Andoni et al., 2015; Xu & Pham, 2024). Beyond standalone similarity estimation, CEOs has also been leveraged to accelerate similarity graphs, which constitute one of the most effective structures for ANNS. By swapping the roles of the query and data vectors in the original CEOs formulation, Lu et al. (2024) developed a space-partitioning technique and proposed the PEOs test. This test enables probabilistic comparisons between the objective angle and a fixed threshold, and has been integrated into the routing procedures of similarity graphs. Specifically, for every visited node u, we send all the out-neighbors of u to PEOs. Only the exact distances between q and the nodes that pass the PEOs test are computed. In the experiments of (Lu et al., 2024), only around 25% nodes can pass the PEOs test. As a result, substantial improvements in search performance brought by PEOs were reported over similarity graphs such as HNSW (Malkov & Yashunin, 2020) and NSSG (Fu et al., 2022). More recently, Lu et al. (2025) proposed a new test function, KS2, which achieves higher test accuracy while maintaining a shorter test time compared with PEOs. KS2 employs a projection structure similar to that of PEOs, but additionally incorporates the reference angle into the testing procedure. Lu et al. (2025) further showed that, without introducing additional assumptions, the test guarantees a success probability of at least 0.5 when deciding whether the exact distance between the objective node and q should be evaluated.
+
+*Table 4.* PAG parameter settings.
+
+<span id="page-17-2"></span>
+
+| Dataset                    | PAG-Base $(efC, M, L)$            | PAG-Lite $(efC, M, L)$          |  |  |  |  |  |
+|----------------------------|-----------------------------------|---------------------------------|--|--|--|--|--|
+| Modern Datasets            |                                   |                                 |  |  |  |  |  |
+| DBpedia1536<br>DBpedia3072 | (1000, 32, 128)<br>(1000, 32, 64) | (100, 32, 128)<br>(100, 32, 64) |  |  |  |  |  |
+| WoltFood                   | (2000, 128, 64)                   | (100, 32, 04)<br>(100, 64, 32)  |  |  |  |  |  |
+| DataCompDr<br>AmazonBooks  | (1000, 32, 64)<br>(2000, 64, 64)  | (100, 32, 64)<br>(100, 64, 32)  |  |  |  |  |  |
+| MajorTOM                   | (1000, 32, 64)                    | (100, 04, 32) $(100, 16, 64)$   |  |  |  |  |  |
+| Legacy Datasets            |                                   |                                 |  |  |  |  |  |
+| Word2Vec                   | (8000, 64, 32)                    | (200, 64, 32)                   |  |  |  |  |  |
+| GIST                       | (1000, 64, 96)                    | (64, 32, 96)                    |  |  |  |  |  |
+| GloVe                      | (2000, 64, 32)                    | (100, 32, 32)                   |  |  |  |  |  |
+| ImageNet                   | (2000, 64, 16)                    | (200, 16, 16)                   |  |  |  |  |  |
+| SIFT10M                    | (1000, 32, 16)                    | (100, 16, 16)                   |  |  |  |  |  |
+| DEEP100M                   | (1000, 32, 16)                    | (100, 16, 16)                   |  |  |  |  |  |
+
+### <span id="page-17-0"></span>**D. Experimental Setup Details**
+
+#### **D.1. Environment**
+
+All experiments were conducted on a machine equipped with an Intel Xeon Platinum 8276L CPU, which supports AVX-512 instructions and provides 112 hardware threads. The system was configured with 754 GB of DDR4 ECC memory and runs Ubuntu 22.04. For indexing, all 112 threads were used, while search was performed using a single CPU thread, in line with the standard setting in ANN-Benchmarks (Bernhardsson et al., 2015).
+
+#### D.2. Baselines
+
+#### D.2.1. SELECTION OF BASELINES
+
+For quantization-based methods, we choose IVF-PQFS (Jégou et al., 2011), ScaNN (Guo et al., 2020a), and RabitQ+ (Gao et al., 2025) as baselines, where RabitQ+ (Gao et al., 2025) is an improved version of RabitQ (Gao & Long, 2024). For graph-based methods, we choose HNSW (Malkov & Yashunin, 2020), Vamana (Subramanya et al., 2019), and SymphonyQG (Gou et al., 2025), where SymphonyQG has shown its superiority over LVQ (Aguerrebere et al., 2023) and NGT-QG (Yahoo! Japan, 2023). On the other hand, HNSW+KS2 (Lu et al., 2025) has been shown to perform better than HNSW+PEOs (Lu et al., 2024), and KS2 can be approximately viewed as our PRT without TFB, modulo threshold-setting differences. Thus, for a fair comparison, we re-implement the KS2 component within our framework and report results for PAG versus PRT only in our ablation study.
+
+### D.2.2. PARAMETER SETTING
+
+For graph-based methods, we used large graph construction parameters to ensure their best search performance. To ensure a fair comparison of indexing time, the efC parameter in PAG-Base was set to a similar or even larger value than HNSW.
+
+{18}------------------------------------------------
+
+<span id="page-18-0"></span>![](_page_18_Figure_1.jpeg)
+
+<span id="page-18-1"></span>Figure 12. QPS-recall of PAG-Base under varying space partition size L, K = 100. M and efC values are given in Table 4, PAG-Base.
+
+![](_page_18_Figure_3.jpeg)
+
+Figure 13. Indexing time and peak memory usage of PAG-Base under varying space partition size  $L.\ M$  and efC values are given in Table 4, PAG-Base.
+
+<span id="page-18-2"></span>![](_page_18_Figure_5.jpeg)
+
+Figure 14. QPS-recall of PAG-Base under varying (M, efC), K = 100. L values are given in Table 4, PAG-Base.
+
+![](_page_18_Figure_7.jpeg)
+
+Figure 15. Indexing time and peak memory usage of PAG-Base under varying (M, efC). L values are given in Table 4, PAG-Base.
+
+(1) **HNSW**: efC=1024. M=32. The only exception is MajorTOM, where efC=512 due to the long indexing time on this dataset.
+
+<span id="page-18-3"></span>
+
+- (2) **Vamana**: By default, R=64. L=1024.  $\alpha=1.2$ . For better QPS-recall performance on MajorTOM and DEEP100M, we choose R=100 and L=250 on MajorTOM, and R=110 and L=200 on DEEP100M.
+- (3) **SymQG**: efC = 1024. There are two options for the degree, 32 and 64, resulting in two baselines, SymQG (32)
+
+- and SymQG (64).
+- (4) **ScaNN**: We adopt the recommended settings from its GitHub repository (Guo et al., 2020b).
+- (5) **RabitQ+**: Following the same setting as in (Gao et al., 2025),  $b=8.\ k=16,384$  for DataCompDr and 4096 for the other datasets.
+- (6) **IVFPQFS**: We test combinations of  $nlist \in \{1024, 4096, 16384\}, k_-factor \in \{64, 128, 256\}, and$
+
+{19}------------------------------------------------
+
+M candidates ∈ [48, 384]. A combination is chosen for good QPS-recall performance on each dataset.
+
+(7) PAG: PAG has two versions, PAG-Base and PAG-Lite. Their parameter settings are listed in Table [4.](#page-17-2) Users can adjust the values of efC, M, and L by taking into account the indexing time, memory footprint, and search efficiency.
+
+other three datasets. PAG-Base also uses less memory than SymQG. The QPS-recall performance of PAG-Lite is generally better than HNSW and Vamana. Meanwhile, PAG-Lite achieves the smallest indexing time, and is competitive in memory footprint.
+
+## <span id="page-19-0"></span>E. Additional Experimental Results
+
+## E.1. Effect of Space Partition Size L
+
+1054
+
+1056
+
+1074 1075 1076
+
+From the QPS-recall results in Fig. [12,](#page-18-0) we can see that, within a moderate range, increasing L leads to better search performance, at the cost of a higher memory footprint. As shown in Fig. [13](#page-18-1) (left), larger L values also results in longer indexing time, due to the faster search speed for each inserted vector. On the other hand, this causes a moderately larger index size, which is reflected in the memory footprint reported in Fig. [13](#page-18-1) (middle and right). As analyzed in Appendix [B,](#page-14-0) L = √ d is an appropriate choice.
+
+### E.2. Effect of M and efC
+
+We vary parameters M and efC and plot the QPS-recall in Fig. [14.](#page-18-2) Larger M and efC value result in faster query processing speed, and the advantage is more remarkable when users require high recall values. As shown in Fig. [15,](#page-18-3) increasing M and efC generally leads to more indexing time as well as larger memory footprint. This is expected, because they control the out-degree and the number of nodes visited for each insertion. The only exception is DataCompDr, where a largerM does not necessarily mean a slower indexing speed. This is because the search speed for each inserted vector is accelerated despite a larger out-degree.
+
+### <span id="page-19-1"></span>E.3. Evaluation on Additional Legacy Datasets
+
+We report the results on four additional legacy datasets, Word2Vec, ImageNet, GIST, and SIFT10M, which are commonly used for ANNS evaluation. Figs. [16,](#page-20-0) [17,](#page-20-2) and [18](#page-20-1) show the QPS-recall performances when K = 10, 100, and 1000. Fig. [19](#page-20-3) shows the indexing time and memory footprint. From the results in figures, we have the following observations. PAG-Base maintains its competitiveness in QPS-recall performance, as we have witnessed in the experiments on other datasets. On Word2Vec and ImageNet, where graph-based methods and SymQG struggle to achieve high recalls, PAG-Base has a significant advantage in QPS. On GIST and SIFT10M, SymQG performs better than PAG-Base in QPS when K = 10 and K = 100, because the two datasets are sparse and well-suited for vector quantization. When K = 1000, PAG-Base outperforms SymQG by a large margin. For indexing speed, PAG-Base is slower than graphbased methods and SymQG on Word2Vec but is faster on the
+
+{20}------------------------------------------------
+
+<span id="page-20-3"></span>
+
+<span id="page-20-0"></span>![](_page_20_Figure_1.jpeg)
+
+Figure 16. QPS-recall on additional legacy datasets, K = 10.
+
+<span id="page-20-2"></span>![](_page_20_Figure_3.jpeg)
+
+Figure 17. QPS-recall on additional legacy datasets, K = 100.
+
+<span id="page-20-1"></span>![](_page_20_Figure_5.jpeg)
+
+Figure 18. QPS-recall on additional legacy datasets, K = 1000.
+
+![](_page_20_Figure_7.jpeg)
+
+Figure 19. Indexing time and peak memory usage on additional legacy datasets.

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+{"paper_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a", "chunk_id": "fddf30e3-e5ae-4a68-b862-daa6e531883a:0134", "section": "References", "page_start": 12, "page_end": 12, "type": "ListGroup", "text": "Bernhardsson, E. Annoy. 2013. Bernhardsson, E., Aumüller, M., and Faithfull, A. Ann benchmarks. 2015. Beygelzimer, A., Kakade, S. M., and Langford, J. Cover trees for nearest neighbor. In ICML , volume 148, pp. 97–104, 2006. Charikar, M. Similarity estimation techniques from rounding algorithms. In STOC , pp. 380–388, 2002. Chen, P. H., Chang, W., Jiang, J., Yu, H., Dhillon, I. S., and Hsieh, C. FINGER: fast inference for graph-based approximate nearest neighbor search. In WWW , pp. 3225–3235, 2023. Chen, T., Fu, C., Wu, J., Wu, H., Fan, H., Ke, X., Gao, Y., Ni, Y., and Zeng, A. Reveal hidden pitfalls and navigate next generation of vector similarity search from task-centric views. arXiv preprint arXiv:2512.12980 , 2025. URL Cheng, R., Peng, Y., Wei, X., Xie, H., Chen, R., Shen, S., and Chen, H. Characterizing the dilemma of performance and index size in billion-scale vector search and breaking it with second-tier memory. arXiv preprint arXiv:2405.03267 , 2024. URL Fensore, C., Dhole, K. D., Ho, J. C., and Agichtein, E. Evaluating hybrid retrieval augmented generation using dynamic test sets: Liverag challenge. CoRR , abs/2506.22644, 2025. URL Fu, C., Xiang, C., Wang, C., and Cai, D. Fast approximate nearest neighbor search with the navigating spreading-out graph. PVLDB , 12(5):461–474, 2019. Fu, C., Wang, C., and Cai, D. High dimensional similarity search with satellite system graph: Efficiency, scalability, and unindexed query compatibility. IEEE Trans. Pattern Anal. Mach. Intell. , 44(8):4139–4150, 2022.", "source": "marker_v2", "marker_block_id": "/page/11/ListGroup/423"}
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+10<sup>2</sup>
+
+[p. 11 | section: References | type: Text]
+10
+
+[p. 11 | section: References | type: Text]
+10
+
+[p. 11 | section: References | type: Text]
+10
+
+[p. 11 | section: References | type: Text]
+10
+
+[p. 11 | section: References | type: Text]
+10<sup>1</sup>
+
+[p. 11 | section: References | type: Text]
+Recall (%) WoltFood (K=800)
+
+[p. 11 | section: References | type: Text]
+AmazonBooks (K=800)
+
+[p. 11 | section: References | type: Text]
+Recall (%) DEEP100M (K=800)
+
+[p. 11 | section: References | type: FigureGroup]
+Figure 7. QPS-recall comparison of PAG-Base and SymQG under varying retrieval size K.
+
+[p. 11 | section: References | type: FigureGroup]
+Figure 8. QPS-recall of query workloads with online insertions.
+
+[p. 11 | section: References | type: FigureGroup]
+Figure 9. Ablation study, QPS-recall. PRT only is the re-implementation of KS2 in PAG.
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+Figure 10. Ablation study, indexing time. PRT only is the reimplementation of KS2 in PAG.
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+Zhang, S., Wang, J., Zhou, R., Liao, J., Feng, Y., Zhang, W., Wen, Y., Li, Z., Xiong, F., Qi, Y., et al. MemRL: Selfevolving agents via runtime reinforcement learning on episodic memory. arXiv preprint arXiv:2601.03192 , 2026. URL .

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+{0}
+## Abstract
+Approximate Nearest Neighbor Search (ANNS) is fundamental to modern AI applications. Most existing solutions optimize query efficiency but fail to align with the practical requirements of modern workloads. In this paper, we outline six critical demands of modern AI applications: high query efficiency, fast indexing, low memory footprint, scalability to high dimensionality, robustness across varying retrieval sizes, and support for online insertions. To satisfy all these demands, we introduce Projection-Augmented Graph (PAG), a new ANNS framework that integrates projection techniques into a graph index. PAG reduces unnecessary exact distance computations through asymmetric comparisons between exact and approximate distances as guided by projection-based statistical tests. Three key components are designed and unified to the graph index to optimize indexing and searching. Experiments on six modern datasets demonstrate that PAG consistently achieves superior query per second (QPS) recall performance—up to 5×faster than HNSW while offering fast indexing speed and moderate memory footprint. PAG remains robust as dimensionality and retrieval size increase and naturally supports online insertions. Our source code is available at: [ [KejingLu-810/PAG/]( .
+## 1. Introduction
+Given a dataset D ⊂ R <sup>d</sup> with size n and a query q ∈ R d , Approximate Nearest Neighbor Search (ANNS) aims to find K approximate nearest neighbors of q as accurately and efficiently as possible. Due to its crucial role in many applications such as image search, recommender systems, and
+retrieval-augmented-generation (RAG), we have witnessed the rapid proliferation of ANNS solvers. Whereas the ANN-Benchmarks [\(Bernhardsson et al.,](#page-11-0) [2015\)](#page-11-0) has been developed to compare these methods in a unified environment, many datasets in the ANN-Benchmarks, as pointed out by [Chen](#page-11-1) [et al.](#page-11-1) [\(2025\)](#page-11-1), are derived from outdated models such as SIFT and GIST, which are image descriptors developed more than 20 years ago [\(Lowe,](#page-12-0) [2004;](#page-12-0) [Oliva & Torralba,](#page-12-1) [2001\)](#page-12-1). In addition, its evaluation metric is limited to query per second (QPS)-recall under K = 10.
+Seeing the fast development of AI technology, we argue that a well-designed ANNS solver should perform well on modern datasets, and beyond (D1) QPS-recall performance, several demands must also be considered: (D2) indexing time, for which graph indexes such as HNSW [\(Malkov & Yashunin,](#page-12-2) [2020\)](#page-12-2) are often criticized [\(Lin,](#page-12-3) [2025;](#page-12-3) [Li & Papakonstantinou,](#page-12-4) [2025\)](#page-12-4), compromising their use in applications that require instant deployment; (D3) memory footprint, where moderate and adjustable consumption, mostly caused by the index size, is desired for the memory vs accuracy trade-off [\(Cheng](#page-11-2) [et al.,](#page-11-2) [2024;](#page-11-2) [Wallace,](#page-13-0) [2025\)](#page-13-0); (D4) scalability to high dimensionality, as motivated by the increasing dimensionality of modern embedding models such as CLIP [\(Radford et al.,](#page-12-5) [2021\)](#page-12-5) [1](#page-0-0) ; (D5) robustness against the retrieval size K, due to the needs in various applications (e.g., K is typically 10 for RAG [\(Fensore et al.,](#page-11-3) [2025;](#page-11-3) [Ke et al.,](#page-11-4) [2025\)](#page-11-4) but can be up to hundreds in image retrieval [\(Vendrow et al.,](#page-13-1) [2024\)](#page-13-1) and thousands in recommender systems [\(Zhang et al.,](#page-13-2) [2025\)](#page-13-2)); (D6) support of online insertions, which is essential for emergent applications such as self-evolving agents that continually accumulate and reuse experience through interaction [\(Ouyang](#page-12-6) [et al.,](#page-12-6) [2025;](#page-12-6) [Zhai et al.,](#page-13-3) [2025;](#page-13-3) [Zhang et al.,](#page-13-4) [2026\)](#page-13-4).
+## <span id="page-0-1"></span>1.1. Prior Works
+We review representative ANNS solvers based on D1 – D6. More discussions on literature can be found in Appendix [C.](#page-15-0)
+<span id="page-0-0"></span><sup>1</sup>Despite the availability of dimensionality reduction techniques, [Weller et al.](#page-13-5) [\(2025\)](#page-13-5) proved that for embedding-based retrieval with dimensionality d, there exists a dataset size n (e.g., n ≈ 4M for d = 1024) beyond which it is theoretically impossible to represent all possible top-2 document combinations, showcasing the necessity for higher dimensionality on large datasets.
+{1}------------------------------------------------
+<span id="page-1-0"></span>Table 1. Summary of empirical performance comparison of notable ANNS solvers, HNSW (Malkov & Yashunin, 2020), Vamana (in-memory DiskANN (Subramanya et al., 2019)), IVFPQFS (fast scan IVFPQ (Jégou et al., 2011)), ScaNN (Guo et al., 2020a), RaBitQ+ (Gao et al., 2025), SymQG (Gou et al., 2025), HNSW+KS2 (Lu et al., 2025), and our solutions PAG-Base (for high QPS) and PAG-Lite (for fast indexing and small index size). Detailed results are available in Sec. 5.2. The evaluation of D1 – D3 is clustered into tiers, where Tier-4 is the best. D4 – D6 are evaluated in yes/no (√/X).
+| | Graph-Based | | Quantization-Based | | | QG | PG | PA | ıG |
+|--------------------------------------------------|-------------|--------|----------------------|--------------------|--------------|--------|----------|----------|----------|
+| Criteria | HNSW | Vamana | IVFPQFS <sup>1</sup> | ScaNN <sup>1</sup> | RaBitQ+ | SymQG | HNSW+KS2 | PAG-Base | PAG-Lite |
+| D1. QPS-recall | Tier-2 | Tier-2 | Tier-1 | Tier-1 | Tier-1 | Tier-3 | Tier-3 | Tier-4 | Tier-3 |
+| <b>D2.</b> Indexing time | Tier-1 | Tier-1 | Tier-3 | Tier-3 | Tier-3 | Tier-2 | Tier-1 | Tier-2 | Tier-4 |
+| <b>D3.</b> Memory footprint <sup>2</sup> | Tier-3 | Tier-3 | Tier-4 | Tier-4 | Tier-4 | Tier-1 | Tier-2 | Tier-3 | Tier-4 |
+| <b>D4.</b> High-dim. scalability | ✓ | ✓ | ✓ | ✓ | ✓ | X | ✓ | ✓ | ✓ |
+| <b>D5.</b> Retrieval size robustness | ✓ | ✓ | ✓ | ✓ | ✓ | Х | ✓ | ✓ | ✓ |
+| <b>D6.</b> Online insertion support <sup>3</sup> | ✓ | ✓ | ✓ | ✓ | $\checkmark$ | X | × | ✓ | ✓ |
+Reranking is enabled for IVFPOFS and ScaNN.
+Graph-Based Methods. These methods construct a similarity graph connecting nearby vectors and hop towards the neighborhood of q. Notable methods are HNSW (Malkov & Yashunin, 2020), NSG (Fu et al., 2019), and DiskANN (Subramanya et al., 2019). They are generally competitive in QPS-recall but slow in building indexes.
+Quantization-Based Methods. These methods compress vectors and rank them using approximate (i.e., quantized) distances to q in pursuit of efficiency. Representative methods are IVFPQ (Jégou et al., 2011) and ScaNN (Guo et al., 2020a). They are fast in building indexes and save memory, but their QPS-recall is generally inferior to graph-based methods.
+Projection-Based Methods. These methods use random (e.g., E2LSH (Andoni & Indyk, 2005), Falconn (Andoni et al., 2015), and CEOs (Pham, 2021)) or data-dependent (e.g., learning to hash (Wang et al., 2018)) projections of the original vectors for indexing. Although many of them enjoy theoretical guarantees, they are less competitive than graphand quantization-based methods in QPS-recall performance, hence becoming less popular in modern top-K ANNS.
+<u>Tree-Based Methods.</u> They utilize tree indexes. table methods include k-d tree (Bentley, 1990), cover tree (Beygelzimer et al., 2006), and Annoy (Bernhardsson, 2013). Like projection-based methods, they are less widely used for modern ANNS due to limited QPS-recall performance.
+As discussed above, graph-based methods are good at QPSrecall. But they compute exact distances between vectors, which is computationally costly. To address this weakness, recent advancements attempt to integrate other techniques, in particular, quantization or projection, into graphs.
+Quantized Graph (QG) Methods. They construct a similarity graph based on quantized vectors instead of the original ones, thereby replacing exact distances with approximate values. Representative methods are NGT-QG (Yahoo! Japan, 2023), LVQ (Aguerrebere et al., 2023), and SymphonyQG (Gou et al., 2025). QG methods can achieve very high QPS-recall performance, yet they are sensitive to the data distribution and many of them do not perform very well in D3 - D6.
+*Projection* + *Graph* (*PG*) *Methods*. Unlike QG, they operate on original vectors and reduce unnecessary exact distance computations. To this end, they employ projection techniques to test whether a neighbor needs to be explored (called routing tests). Representative methods are FINGER (Chen et al., 2023), PEOs (Lu et al., 2024), and KS2 (Lu et al., 2025). Despite QPS-recall improvement on top of graph-based methods, such improvement comes at the cost of indexing time, memory footprint, and support for online insertions.
+Table 1 compares notable methods implemented in leading vector databases and recent ones representing state of the art.
+#### 1.2. Our Solution
+We propose a new ANNS framework, Projection-Augmented Graph (PAG), which integrates projection into a similarity graph and achieves superior performance for all the six criteria. Unlike PG methods, PAG treats projection as a fundamental building block of graph construction rather than as a plug-in. The rationale of PAG is to accommodate both exact and approximate distances within a *unified* framework, addressing two key issues: (1) when distances shall be computed exactly and when they shall be approximated, and (2) how exact and approximate distances are compared during indexing and searching to enhance the performance.
+PAG significantly reduces the computational cost of searching (D1) and indexing (D2) by carefully determining whether exact distance computation is needed. Such procedure relies on asymmetric comparisons between exact and approximate distance values obtained from space-efficient and adjustable random-projection structures (D3), as opposed to symmetric comparisons in QG. Moreover, the asymmetric distance
+Memory footprint for searching, rather than indexing, is compared.
+<sup>&</sup>lt;sup>3</sup> For any streaming workload Q composed of search and insertion queries, we say an algorithm supports online insertion, if the amortized cost of processing a query in Q is O(1) times the cost of a search. In other words, the index can be incrementally updated with minimal cost. See Sec. 4.3 for the analysis of PAG.
+{2}------------------------------------------------
+comparisons, as guided by random-projection-based statistical tests, only need to tell which value is larger, thereby achieving accuracy while preserving efficiency. Meanwhile, PAG inherits graph-based methods' scalability to high dimensionality (D4) and robustness against retrieval size (D5), and follows the search-and-insertion paradigm (e.g., HNSW) to accommodate online insertions (D6).
+Our technical contributions are summarized as follows.
+- (1) Following the idea of routing tests in PEOs [\(Lu et al.,](#page-12-10) [2024\)](#page-12-10) and KS2 [\(Lu et al.,](#page-12-8) [2025\)](#page-12-8), we derive a Probabilistic Routing Test (PRT) function (Sec. [3.2\)](#page-4-0). The theoretical result (Theorem [3.1\)](#page-4-1), along with Lemma 4.3 in [Lu et al.](#page-12-8) [\(2025\)](#page-12-8), provides a complete theoretical explanation of PRT. In addition, our work is the first to apply PRT to graph construction.
+- (2) We propose a data structure called Test Feedback Buffer (TFB) (Sec. [3.3\)](#page-5-0), which refines the threshold setting in PRT and enables the reuse of false positives generated by PRT. By incorporating TFB into PRT, we obtain the PRT-TFB test as a core technique for accelerating both indexing and searching.
+- (3) We propose a statistical test called Probabilistic Edge Selection (PES) (Sec. [3.4\)](#page-5-1), which is derived from Theorem [3.1](#page-4-1) and can expand in-degrees when necessary. In collaboration with PRT, PRT-PES can improve the search performance for hard datasets on which traditional graph indexes perform poorly, while incurring very small indexing overhead.
+- (4) PRT, TFB, and PES constitute PAG (Fig. [1\)](#page-3-0). We show how the three components interact within PAG, with implementation details and complexity analysis provided.
+- (5) We conduct experiments on six modern (post-2023) datasets covering text, image, and multimodal data, with dimensionality ranging from 384 to 3072 and retrieval size from 10 to 1000. The results show that PAG achieves the best QPS-recall performance (up to 5× faster than HNSW), and its superiority is particularly evident on datasets with higher dimensionality. PAG remains dominant as K increases. By adjusting parameters, PAG can deliver the fastest indexing speed and lowest memory footprint on most datasets while maintaining competitive QPS-recall. PAG also performs well on the datasets in the ANN-Benchmarks, showcasing its compatibility with data generated by legacy models.
+## 2. Problem Setting
+Since PAG adopts a search-and-insertion paradigm, its search process, similar to HNSW, can be seen as a special case of index construction (i.e., no update to the graph), and we can focus on the problems that arise in index construction. Because constructing a similarity graph is essentially determining its edge set, we consider the following question: *Can we design statistical methods that select edges efficiently (i.e., fast indexing) and effectively (i.e., fast searching)?* To answer this question, we first review the existing edge-selection strategy, and then formalize the problems to be solved.
+#### 2.1. Background of RobustPrune
+Given a directed graph, let v be a node whose edges are to be determined. To identify its in-neighbor set Nin(v) and out-neighbor set Nout(v), state-of-the-art methods, such as HNSW, NSG and Vamana, employ a strategy known as RobustPrune to decide whether a candidate edge should be preserved. Take the construction of Nin(v) as an example. For a candidate neighbor u of node v, the pruning criterion considers the following set [2](#page-2-0) :
+$$S_{u,v} = \{ w \in N_{\text{out}}(u) \mid ||w - v|| \le ||v - u|| \}.$$
+(1)
+Nout(v)*.* We take v as the query and conduct an ANNS from v to obtain a candidate set, which is stored in a priority queue P. RobustPrune is then applied to P to determine Nout(v).
+Nin(v)*.* We check every u in Nout(v) and preserve those u's retained by RobustPrune as Nin(v).
+### 2.2. Towards Fast Graph Construction
+During graph construction, most of the time is spent on ANNS for the candidate set of Nout(v). Although we can reduce the graph construction parameter (e.g., efC in HNSW) for fast graph construction, the query processing performance degrades accordingly. Thus, we raise the following question.
+Q1. *How can we accelerate the ANNS for* v *while preserving the query processing performance?*
+Existing studies PEOs and KS2 have shown that PRT has the potential to be an appropriate answer to Q1. For each visited node u and its out-neighbor w, PRT sends (u, v, w) to a probabilistic test function to see whether they can pass the test, and then determines whether the exact distance between v and w needs to be computed based on the test result. Consequently, PRT avoids unnecessary distance computations, leading to faster search. In KS2 [\(Lu et al.,](#page-12-8) [2025\)](#page-12-8), it is shown that the routing test under ℓ<sup>2</sup> and cosine distances is equivalent to the angle-thresholding problem when vector norms
+<span id="page-2-0"></span>For Vamana, we assume its pruning ratio α = 1 and omit it in the inequality.
+{3}------------------------------------------------
+<span id="page-3-0"></span>![](_page_3_Figure_2.jpeg)
+Figure 1. An overview of PAG. In this example, $\boldsymbol{u}$ has 7 out-neighbors $\{\boldsymbol{w_i}\}_{i=1}^7$ . Let $\{\boldsymbol{e_i}\}_{i=1}^7$ denote the edges between $\boldsymbol{u}$ and $\{\boldsymbol{w_i}\}_{i=1}^7$ , which are sent to PRT, where the threshold is determined by $\boldsymbol{z_{max}}$ . As a result, only $\boldsymbol{w_3}$ and $\boldsymbol{w_5}$ pass PRT, and their exact distances to $\boldsymbol{v}$ are computed. By distance comparison, node $\boldsymbol{w_5}$ is identified as a false positive, not added to W, and thus sent to $R_F$ . Node $\boldsymbol{w_3}$ is inserted into W, causing $\boldsymbol{z_{max}}$ to be ejected from W and sent to $R_T$ . The two rings $R_F$ and $R_T$ are merged and refilled into W. On the other route, by a left shift, the signs of both $\boldsymbol{w_3}$ and $\boldsymbol{w_5}$ are reversed, and all signs become negative, indicating that $\boldsymbol{u}$ is rejected by the PES test. Consequently, $\boldsymbol{u}\dot{\boldsymbol{v}}$ is treated as a candidate edge and added to the PES set.
+are stored explicitly. Therefore, PRT can be formulated as follows:
+<span id="page-3-2"></span>**Problem 2.1.** (**Probabilistic Routing Test**) Given $u, v, w \in \mathbb{S}^{d-1}$ and a threshold $-1 \le \tau \le 1$ , construct a random-projection vector set $\mathcal{F}$ and design a test function w.r.t $\mathcal{F}$ , i.e., PRT : $(u, v, w, \tau) \to RV$ with time complexity o(d), where RV denotes the set of one-dimensional random variables, such that if $\cos(\angle(v-u, w-u)) \ge \tau$ , then $\mathbb{P}[\operatorname{PRT}(u, v, w, \tau) \ge 0] \ge 0.5$ . Otherwise, $\mathbb{P}[\operatorname{PRT}(u, v, w, \tau) < 0] \to 1$ as $|\mathcal{F}| \to \infty$ .
+Different from the routing test functions used in PEOs and KS2, we take $\tau$ as an explicit input. In this paper, we use another method to determine $\tau$ in order to further improve the efficiency of the routing test.
+#### 2.3. Towards High Graph Connectivity
+From the process of determining $N_{\rm in}(\boldsymbol{v})$ , we observe that its candidate set is restricted to $N_{\rm out}(\boldsymbol{v})$ . Recent work (Wang et al., 2025a) shows that $|N_{\rm out}(\boldsymbol{v})|$ may be too small to reliably determine incoming edges, causing some nodes to have very small in-degrees. Such nodes may become unreachable during search, which partly explains why HNSW performs poorly on certain real-world datasets. On the other hand, for each checked node $\boldsymbol{u}$ , the worst-case time complexity of RobustPrune can reach $O(|N_{\rm out}(\boldsymbol{u})|d)$ , implying that a straightforward enlargement of the candidate set for
+<span id="page-3-1"></span>![](_page_3_Figure_9.jpeg)
+Figure 2. Illustration of PES (left: the role of PES; right: geometric illustration). Let $\boldsymbol{v}$ be the node to be inserted. The four green nodes are out-neighbors of $\boldsymbol{v}$ . Without PES, we can obtain only a single in-neighbor via RobustPrune. By taking all other visited nodes (the blue ones) into account, we apply PES, followed by RobustPrune. As such, we can identify three additional promising in-neighbors, thereby strengthening the connectivity of the neighborhood of $\boldsymbol{v}$ .
+determining incoming edges is expensive. Considering this dilemma, we raise the following question.
+Q2. How can we efficiently detect incoming edges outside $N_{\text{out}}(v)$ that are useful for improving query processing performance but hard to be identified by RobustPrune?
+{4}------------------------------------------------
+To answer this question, we formulate a problem as follows.
+<span id="page-4-2"></span>**Problem 2.2.** (**Probabilistic Edge Selection**) Given $u, v \in \mathbb{S}^{d-1}$ , construct a random-projection-vector set $\mathcal{F}$ and design a probabilistic edge selection function PES: $(u, v) \to RV$ whose time complexity is $O(|N_{\mathrm{out}}(u)|)$ , such that if $S_{u,v}$ is non-empty, then $\mathbb{P}[\mathrm{PES}(u,v) \geq 0] \geq 0.5$ . Otherwise, $\mathbb{P}[\mathrm{PES}(u,v) < 0] \to 1$ as $|\mathcal{F}| \to \infty$ .
+We emphasize that the PES test is far more than a fast probabilistic implementation of RobustPrune. This is because the PES test can be applied to all visited nodes during the ANNS of $\boldsymbol{v}$ , whose number is typically much larger than $N_{\rm out}(\boldsymbol{v})$ in practice, leading to a better graph index in terms of connectivity (Fig. 2).
+## 3. Projection-Augmented Graph
+We show that Problems 2.1 and 2.2 can be solved in a unified framework PAG, which contains three components: Probabilistic Routing Test (PRT), Test Feedback Buffer (TFB), and Probabilistic Edge Selection (PES).
+### <span id="page-4-4"></span>3.1. Neighborhood Relations via Random Projection
+We present an asymptotic result that characterizes the relationship between multiple angles in high-dimensional spaces and their corresponding projection values onto a certain projection vector. This result forms the theoretical basis for all the three components. We first describe how to construct the random-projection vector set $\mathcal F$ that appears in Problems 2.1 and 2.2, following an approach similar to that used in KS2 (Lu et al., 2025). Specifically, we divide the original space $\mathbb R^d$ into L subspaces, each of dimension d/L. In each subspace, we generate multiple cross-polytopes and apply an independent rotation to each, producing a total of m normalized vectors on $\mathbb S^{d/L-1}$ . By concatenation, we obtain a total of $m^L$ normalized vectors $\{r_j\}_{j=1}^{m^L}$ , which together form the set $\mathcal F$ . Consider v as the node to be inserted and v as the candidate neighbor to be checked, let $N_{\mathrm{out}}(u) := \{w_i\}_{i=1}^t$ . We define $\{\alpha_i\}_{i=1}^t$ as follows.
+<span id="page-4-3"></span>
+$$\alpha_i := \arccos \frac{\langle \boldsymbol{w_i} - \boldsymbol{u}, \boldsymbol{v} - \boldsymbol{u} \rangle}{\|\boldsymbol{w_i} - \boldsymbol{u}\| \|\boldsymbol{v} - \boldsymbol{u}\|}, \quad 1 \le i \le t \quad (2)$$
+Since $\|\boldsymbol{w_i} - \boldsymbol{u}\|$ and $\|\boldsymbol{v} - \boldsymbol{u}\|$ can be pre-computed, we aim to estimate $[\cos \alpha_1, \dots, \cos \alpha_t]^{\top}$ without exact inner product computation in Eq. (2). We will show that this can be realized by $\mathcal{F}$ . For each $\boldsymbol{w_i}$ ( $1 \leq i \leq t$ ), let $\boldsymbol{r_i^*} \in \mathcal{F}$ be the reference vector that has the smallest angle with $\boldsymbol{w_i} - \boldsymbol{u}$ , and denote this smallest angle by $\beta_i$ . We use $\cos \theta_i$ to denote the cosine of the angle between $\boldsymbol{r_i^*}$ and $\boldsymbol{v} - \boldsymbol{u}$ , and subscript l to denote the l-th sub-vector of the original vector in $\mathbb{R}^d$ , $1 \leq l \leq L$ . We introduce the following assumptions for each i:
+(A1)
+$$\|(\boldsymbol{w_i} - \boldsymbol{u})\| = 1$$
+and $\|(\boldsymbol{v} - \boldsymbol{u})\| = 1$ .
+(A2)
+$$\|(\boldsymbol{w_i} - \boldsymbol{u})_l\|$$
+and $\|(\boldsymbol{v} - \boldsymbol{u})_l\|$ are equal to $(1 + o(1))/\sqrt{L}$ .
+(A3) $\langle (\boldsymbol{v} - \boldsymbol{u})_l, (\boldsymbol{w_i} - \boldsymbol{u})_l \rangle = (\cos \alpha_i)(1 + o(1))/L$ .
+Here, A1 does not lose generality, and A2 and A3 are mild for large d and L (see the remarks in Appendix B). Under these assumptions, the following theorem shows that $\{\alpha_i\}_{i=1}^t$ can be asymptotically estimated by $\{\beta_i\}_{i=1}^t$ and $\{\theta_i\}_{i=1}^t$ .
+<span id="page-4-1"></span>**Theorem 3.1.** Under A1–A3, as $L \to \infty$ , $d/L \to \infty$ , and m grows sufficiently fast with respect to d/L, $X = [\cos \theta_1, \dots, \cos \theta_t]^\top$ , conditioned on $\{\alpha_i, \beta_i\}_{i=1}^t$ , is asymptotically Gaussian:
+$$X \mid \{\alpha_i, \beta_i\}_{i=1}^t \xrightarrow{d} \mathcal{N}(\bar{\mu}, \bar{\Sigma}_{m,L})$$
+(3)
+where the mean is $\bar{\mu} = [\cos \alpha_1 \cos \beta_1, \dots, \cos \alpha_t \cos \beta_t]^\top$ , and $\bar{\Sigma}_{m,L} = O(\epsilon_m/L)$ with $\epsilon_m \to 0$ as $m \to \infty$ .
+**Remarks.** (1) Theorem 3.1 establishes a geometric relationship among multiple out-neighbors of $\boldsymbol{u}$ and $\boldsymbol{v}$ , allowing us to estimate the relative location of $\boldsymbol{v}$ w.r.t. $\boldsymbol{u}$ within the neighborhood of $\boldsymbol{u}$ . This relationship is the key to both PRT and PES. (2) In our proof, $m \to \infty$ is used to eliminate noise from off-diagonal entries, while $L \to \infty$ is required for the central limit theorem (CLT). In practice, moderate values of m and L are sufficient. Weak results under finite (m, L) can be found in Lemma 4.3 in the KS2 paper (Lu et al., 2025). (3) $\boldsymbol{X}$ can be computed efficiently by AVX512 on modern CPUs. Vector $[\cos \beta_1, \ldots, \cos \beta_t]^{\top}$ can be pre-computed, meaning that we can obtain an efficient way to estimate all $\cos \alpha_i$ 's simultaneously.
+#### <span id="page-4-0"></span>3.2. Probabilistic Routing Test
+Let $\tau_i$ be a threshold w.r.t $w_i$ . Our PRT function is as follows:
+<span id="page-4-5"></span>
+$$PRT(\boldsymbol{u}, \boldsymbol{v}, \boldsymbol{w_i}, \tau_i) = \frac{\cos \theta_i}{\cos \beta_i} - \tau_i.$$
+(4)
+For fixed (u, v), if the value of PRT is positive, the corresponding $w_i$ passes the PRT. Based on Theorem 3.1, under asymptotic assumptions, the PRT function satisfies all the required properties in Problem 2.1. Here, we do not specify the setting of $\tau_i$ , which will be postponed to Sec. 3.3.
+**Remarks.** Following the use of routing test in PEOs and KS2, exact distances are computed only between v and those $w_i$ 's that pass PRT (see Fig. 1). The PRT function has the same structure as the KS2 test function (Lu et al., 2025), except for the setting of the threshold $\tau$ . However, since the PRT and KS2 tests are derived using different principles, their theoretical results are complementary. Theorem 4.3 in the KS2 paper shows that, for every single $w_i$ , even for finite (m, L), if $\cos(\angle(v - u, w_i - u)) \ge \tau_i$ , then $\mathbb{P}[\operatorname{PRT}(u, v, w_i, \tau_i) \ge 0] \ge 0.5$ still holds. Our result, on the other hand, characterizes the concrete asymptotic distribution and reveals the impact of L on the covariance, thereby explaining how L is used to adjust the estimation accuracy.
+{5}------------------------------------------------
+## <span id="page-5-0"></span>3.3. Test Feedback Buffer
+328 329 For the PRT function, a key issue is how to set an appropriate threshold τ . In PEOs and KS2, τ is simply determined by the current furthest point in the result list (priority queue) P. However, due to the existence of false positives (FPs) generated by PRT, some points may pass PRT but cannot be added to P, which implies that the exact distance computations w.r.t such points are redundant. On the other hand, the Gaussian distribution established in Theorem [3.1](#page-4-1) implies that, with high probability, the actual distances of these FPs to the query q are not much larger than the threshold τ . This observation naturally raises the following question: *can we incrementally increase the threshold* τ *so that the currently generated FPs can be reused in subsequent search process?*
+We give an affirmative answer to this question and propose TFB, which consists of four components: the top-K result list RL, the working set W, and two ring buffers R<sup>F</sup> and R<sup>T</sup> , where |W| = |R<sup>F</sup> | = |R<sup>T</sup> |. The search procedure is divided into multiple rounds. In the j-th round, we operate on W and adopt the standard Best-First Search (BFS) strategy to update its nodes. During the BFS over W, in addition to the nodes stored in W, there are two types of nodes whose exact distances to the query q are computed. The first type consists of nodes ejected from W; these nodes are inserted into R<sup>T</sup> . The second type consists of false-positive (FP) nodes that pass the PRT but cannot be added to W; these nodes are inserted into R<sup>F</sup> . After all nodes in W have been visited, we update R<sup>L</sup> and clear W. We then merge R<sup>F</sup> and R<sup>T</sup> and sort the combined list by distance to q. The sorted nodes are first inserted into W until it is full, and the remaining nodes are inserted into R<sup>T</sup> . Finally, R<sup>F</sup> is cleared. The ANNS is completed after several such rounds. In the j-th round, let zmax denote the furthest node in W. τ<sup>i</sup> in Eq. [\(4\)](#page-4-5) is set to be the threshold on the cosine of the angle between w<sup>i</sup> − u and v − u required for w<sup>i</sup> to be admitted into W, i.e.,
+<span id="page-5-2"></span>
+$$\tau_i = \frac{\|u - w_i\|^2 + \|u - v\|^2 - \|z_{\max} - v\|^2}{2\|v - u\|\|w_i - u\|}.$$
+(5)
+*PRT-TFB Test.* We call the combination of Eq. [\(4\)](#page-4-5) and Eq. [\(5\)](#page-5-2) the PRT-TFB test. Compared with existing routing test methods such as PEOs and KS2, which operate on the whole priority queue P, the PRT-TFB test has the following two advantages. (1) For large efC or large efS—which denotes the result list size, same as the notation in HNSW—the size of W is much smaller than that of P, and the movement of elements in W is much faster than that in P. (2) Each FP has a good chance of being selected as the visited node in the future rounds only if it remains in either of two rings, ensuring that its exact distance to q can be utilized at a certain time point.
+#### <span id="page-5-1"></span>3.4. Probabilistic Edge Selection
+In RobustPrune, ∥w<sup>i</sup> − v∥ needs to be smaller than ∥v − u∥ to allow u to be retained. Let δ<sup>i</sup> := ∥w<sup>i</sup> − u∥/(2∥v − u∥) denote the threshold on the cosine of the angle between w<sup>i</sup> − u and v − u, such that ∥w<sup>i</sup> − v∥ < ∥v − u∥. Then, the PES function is designed as follows.
+<span id="page-5-3"></span>
+$$PES(\boldsymbol{u}, \boldsymbol{v}) = \max_{1 \le i \le t} \left( \frac{\cos \theta_i}{\cos \beta_i} - \delta_i \right).$$
+(6)
+By Theorem [3.1,](#page-4-1) we can see that the PES function in Eq. [\(6\)](#page-5-3) satisfies the probabilistic properties in Problem [2.2.](#page-4-2) In practice, if the value in Eq. [\(6\)](#page-5-3) is negative, we say that (u, v) is rejected by PES, and this edge will be regarded as a promising candidate for further examination of RobustPrune.
+*PRT-PES Collaboration.* From their definitions, the difference between the two thresholds, that is, τ<sup>i</sup> − δ<sup>i</sup> , is a value independent of v − w and can be pre-computed. Thus, we can conduct the PES test based on the test results of the PRT, leading to an O(1) time complexity, which confirms the objective O(|Nout(u)|) stated in Problem [2.2.](#page-4-2)
+*PES Set.* For every u whose PES value is negative, we do not check uv⃗ by RobustPrune immediately. Instead, we add uv⃗ into a so-called PES set for later examination (see Fig. [1\)](#page-3-0).
+## 4. Implementation and Analysis
+#### 4.1. Implementation
+*Indexing Phase.* PAG inserts every v ∈ D sequentially into the graph. For each v, the process can be divided into the following three steps.
+Step 1. PAG performs ANNS equipped with the PRT-TFB test to obtain the elements stored in RL, and then uses RobustPrune to determine Nout(v) and Nin(v).
+Step 2. Let Cand(v) denote the set of all nodes visited during ANNS. PAG uses PRT-PES to select additional in-neighbors from Cand(v), thereby supplementing the PES set.
+Based on our preceding discussion, PRT-TFB ensures the efficiency of ANNS, while PRT-PES detects more promising edges. We finally select more edges from the PES set.
+Step 3. For each candidate edge in the PES set, we apply RobustPrune to determine if it can be added to the graph.
+*Searching Phase.* The query processing with PAG is exactly the PRT-TFB-based ANNS for a query q.
+*Online Insertion.* Like HNSW, PAG naturally supports online insertion. The PES set is checked only after enough new nodes have been inserted into the graph.
+{6}------------------------------------------------
+#### <span id="page-6-1"></span>330 331 334 336 338 339 340 341 342 343 344 345 346 347 348 349 350 351 354 356 358 359 360 361 362 363 364 365 Algorithm 1: Construction of PAG Input: G := ∅ is the graph to be built; D is the dataset; 2M is the maximum out-degree; L is the space partition size; efC is the maximum size of visited nodes; b is the size of W Output: G with node set D, edge set E, and inference structure I <sup>1</sup> PES set is an empty table to accommodate candidate edges; <sup>2</sup> The capacity of R<sup>L</sup> is set to efC; <sup>3</sup> Maximum round Rmax is set to efC/b; <sup>4</sup> Generate multi-level random-projection structure F = {rjl} (1 ≤ j ≤ m, 1 ≤ l ≤ L); <sup>5</sup> Select b initial nodes, connect them, and write the projection information w.r.t the connected edges to I. Let V denote the set of all initial nodes; <sup>6</sup> foreach v ∈ D\V do <sup>7</sup> Insert b nodes to W and compute their distances to v; <sup>8</sup> Compute all ⟨vl, rjl⟩'s and build a projection table; <sup>9</sup> for r = 1 to Rmax do <sup>10</sup> foreach unvisited u ∈ W do <sup>11</sup> foreach w in Nout(u) do <sup>12</sup> if w passes the PRT-TFB test then <sup>13</sup> Compute the distance between v and w; <sup>14</sup> if ∥w − v∥ < ∥zmax − v∥ then <sup>15</sup> Update W and insert zmax to R<sup>T</sup> ; <sup>16</sup> else <sup>17</sup> Insert w to R<sup>F</sup> ; <sup>18</sup> if (u, v) is rejected by the PRT-PES test then <sup>19</sup> Add (u, v) to the PES set; <sup>20</sup> Update the nodes in W to RL, and empty W; <sup>21</sup> Sort and merge R<sup>F</sup> and R<sup>T</sup> , and empty R<sup>F</sup> and R<sup>T</sup> ; <sup>22</sup> Refill W until it is full using the sorted elements, and refill R<sup>T</sup> with the remaining sorted nodes; <sup>23</sup> Apply RobustPrune to R<sup>L</sup> to compute Nout(v) and Nin(v);
+#### 4.2. Algorithm Pseudo-Codes of PAG
+The pseudo-codes of the PAG algorithms are presented in Algs. [1](#page-6-1) and [2,](#page-6-2) which represent the indexing and searching, respectively. Note that indexing is essentially the sequential insertion of all nodes. Similar to [Lu et al.](#page-12-10) [\(2024\)](#page-12-10), we store the computed projections in a projection table for lookup (Line 8, Alg. [1](#page-6-1) and Line 4, Alg. [2\)](#page-6-2).
+<sup>24</sup> Add the projection information of new edges to I; <sup>25</sup> Apply RobustPrune to the PES set, supplement Nin(v)'s, and add the projection information of new edges to I;
+## <span id="page-6-0"></span>4.3. Complexity Analysis
+*Time complexity.* Searching with HNSW is in O(n ′d)-time, where n ′ is the number of visited nodes during the search. Although a rigorous analysis of n ′ remains an open problem, n ′ can be roughly estimated as O(M log n), where 2M is the
+#### Algorithm 2: Search with PAG
+```
+Input: G(D, E, I) is the constructed similarity graph; F
+is the random-projection structure; efS is the
+result list size; q is the query; K is the retrieval size
+Output: Top-K ANN nodes
+1 The capacity of RL is set to K;
+2 b = max{10, K};
+3 Maximum round Rmax is set to efS/b;
+4 Compute all ⟨ql, rjl⟩'s and build a projection table;
+5 Insert b elements to W and compute their distance to q;
+6 for r = 1 to Rmax do
+7 foreach unvisited u ∈ W do
+8 foreach w in Nout(u) do
+9 if w passes PRT-TFB test then
+10 Compute the distance between q and w;
+11 if ∥w − q∥ < ∥zmax − q∥ then
+12 Update W and insert zmax to RT ;
+13 else
+14 Insert w to RF ;
+15 Update the nodes in W to RL, and empty W;
+16 Sort and merge RF and RT , and empty RF and RT ;
+17 Refill W until it is full using the sorted elements, and
+refill RT with the remaining sorted nodes;
+18 Return the top-K nodes in RL;
+```
+maximum out-degree. In contrast, the complexity of PAG is O(n ′L+γn′d), whereL ≪ d is the user-specified parameter and γ ≤ 0.5 denotes the ratio of nodes that passed the PRT-PES test relative to the total number of candidates checked. γ is small in practice (generally < 0.2), and is smaller than the ratio in PEOs and KS2, because TFB ensures that the threshold increases incrementally rather than being set to the furthest distance w.r.t. a full priority queue.
+As for indexing, due to the search-and-insertion paradigm, we focus on the insertion of v, for which the complexity is O(n ′L + γn′d + M2d + M dm). Here, O(M2d) is the complexity of node connection, and O(M dm) is used to compute projection information, both of which are much smaller than the ANNS part due to their independence of n ′ .
+*Space complexity.* The space complexity of searching with PAG is O(nd + nM + nML), where O(nd) corresponds to the original dataset, O(nM) corresponds to the edge set and O(nML) corresponds to the inference structure. In the indexing phase, since we adopt in-place search for insertion, we only require an additional space of at most O(Mn) to store the edges in PES set. To further reduce the memory footprint, we can follow LVQ [\(Aguerrebere et al.,](#page-9-0) [2023\)](#page-9-0) and represent floats with 2 bytes.
+### 4.4. Parameter Analysis of PAG
+We analyze the settings of the parameters in PAG.
+(1) m: The value of m is fixed to 16, so that each projection vector in a subspace is represented by 4 bits, which is com
+{7}------------------------------------------------
+*Table 2.* Dataset statistics.
+<span id="page-7-1"></span>Name/Source Dataset Size Query Size Dim. OOD Type Embedding Model Distance Measure Modern Datasets DBpedia1536 999,000 1,000 1,536 No Text text-embedding-3-large [\(OpenAI,](#page-12-11) [2024\)](#page-12-11) Euclidean DBpedia3072 999,000 1,000 3,072 No Text text-embedding-3-large [\(OpenAI,](#page-12-11) [2024\)](#page-12-11) Euclidean WoltFood 1,719,611 1,000 512 No Image clip-ViT-B-32 [\(Radford et al.,](#page-12-5) [2021\)](#page-12-5) Euclidean DataCompDr 12,779,520 1,000 1,536 Yes Text-to-Image coca ViT-L-14 [\(Yu et al.,](#page-13-9) [2022\)](#page-13-9) Euclidean AmazonBooks 15,928,208 1,000 384 No Text all-MiniLM-L12-v2 [\(Wang et al.,](#page-13-10) [2020\)](#page-13-10) Euclidean MajorTOM 56,506,400 10,000 1,024 No Image DINOv2 [\(Oquab et al.,](#page-12-12) [2024\)](#page-12-12) Euclidean Legacy Datasets Word2Vec 1,000,000 1,000 300 No Text Word2Vec [\(Mikolov et al.,](#page-12-13) [2013\)](#page-12-13) Euclidean GIST 1,000,000 1,000 960 No Image GIST [\(Oliva & Torralba,](#page-12-1) [2001\)](#page-12-1) Euclidean GloVe 1,193,514 1,000 200 No Text GloVe [\(Pennington et al.,](#page-12-14) [2014\)](#page-12-14) Euclidean ImageNet 2,340,373 200 150 No Image dense SIFT [\(Lazebnik et al.,](#page-11-13) [2006\)](#page-11-13) Euclidean SIFT10M 10,000,000 1,000 128 No Image SIFT [\(Lowe,](#page-12-0) [2004\)](#page-12-0) Cosine DEEP100M 100,000,000 1,000 96 No Image GoogLeNet [\(Szegedy et al.,](#page-12-15) [2015\)](#page-12-15) Cosine
+patible with the AVX512 instruction set. The choice of 4 bits follows NGT-QG [\(Yahoo! Japan,](#page-13-7) [2023\)](#page-13-7).
+- (2) L: L is recommended to be in [8, d/8], where 8 is compatible with AVX512 because at most 8 levels can be accessed at a time, while d/L = 8 is a safe ratio that ensures the accuracy of the multi-level projection structure, as explained in PEOs [\(Lu et al.,](#page-12-10) [2024\)](#page-12-10). Based on the analysis in Sec. [4.3,](#page-6-0) L is used to tune the trade-off between efficiency and accuracy.
+- (3) M: Similar to its usage in HNSW, 2M is the maximum out-degree of PAG. Based on the analysis in Sec. [4.3,](#page-6-0) M is recommended to be 64 when space cost permits. In practice, M is generally chosen from {16, 32, 64}.
+- (4) |W|: The size of W is set to max{10, K}. Under this setting, when K ≥ 10, W is aligned with the result list, which enables refilling to be easily executed.
+- (5) efC: efC determines how many nodes are visited in W and plays a role similar to efC in HNSW. It is used to control the trade-off between indexing time and search performance. For fast indexing, efC is recommended to be in [100, 200]. For high graph quality, efC is recommended to be in [1000, 10000]. Notably, thanks to TFB, PAG with a very large efC can be built much faster than HNSW with the same efC.
+## 5. Experiments
+We report main results here. Detailed setup and more results are available in Appendices [D](#page-17-0) and [E,](#page-19-0) respectively.
+#### 5.1. Experimental Setup
+*Data Statistics.* In the main experiments, we evaluate PAG on eight datasets: DBpedia1536 [\(Qdrant,](#page-12-16) [2024a\)](#page-12-16) (a.k.a., OpenAI-1536), DBpedia3072 [\(Qdrant,](#page-12-17) [2024b\)](#page-12-17) (a.k.a., OpenAI-3072), WoltFood [\(Qdrant,](#page-12-18) [2024c\)](#page-12-18), AmazonBooks [\(Kang et al.,](#page-11-14) [2024\)](#page-11-14), DataCompDr [\(Apple,](#page-10-3) [2025\)](#page-10-3),
+MajorTOM [\(Major TOM,](#page-12-19) [2024\)](#page-12-19), GloVe [\(Pennington et al.,](#page-12-14) [2014\)](#page-12-14), and DEEP100M [\(Simhadri et al.,](#page-12-20) [2021\)](#page-12-20). The first six are modern text, image, and multimodal (text-to-image) datasets generated by recent embedding models. The latter two are widely-used legacy datasets (downloaded from [The Similarity Search Team, CUHK](#page-12-21) [\(2018\)](#page-12-21)). In addition, we report the experimental results on other legacy datasets (Word2Vec, GIST, ImageNet, and SIFT10M) in Appendix [E.3.](#page-19-1) Dataset statistics are reported in Table [2.](#page-7-1) The queries of DataCompDr are out-of-distribution (OOD). Fig. [11](#page-17-1) in Appendix [D](#page-17-0) plots the data and query distributions of this dataset.
+*Baselines.* Baselines include HNSW [\(Malkov & Yashunin,](#page-12-2) [2020\)](#page-12-2), Vamana [\(Subramanya et al.,](#page-12-7) [2019\)](#page-12-7), SymQG [\(Gou](#page-11-8) [et al.,](#page-11-8) [2025\)](#page-11-8), ScaNN [\(Guo et al.,](#page-11-6) [2020a\)](#page-11-6), IVFPQFS [\(Jegou](#page-11-5) ´ [et al.,](#page-11-5) [2011\)](#page-11-5), and RaBitQ+ [\(Gao et al.,](#page-11-7) [2025\)](#page-11-7). We design two variants of PAG by tuning its parameters, as shown in Table [4](#page-17-2) in Appendix [D:](#page-17-0) (1) PAG-Base, for higher search performance, and (2) PAG-Lite, for faster indexing and smaller memory footprint.
+#### <span id="page-7-0"></span>5.2. Experimental Results
+We set the default retrieval size K = 100.
+- *D1. QPS-Recall.* Fig. [3](#page-8-0) plots the QPS-recall performance. PAG-Base performs the best on all the modern datasets except for high recall settings on WoltFood. Its speedup over HNSW can be up to 5 times. For legacy datasets, PAG-Base is the best on GloVe and only second to SymQG on DEEP100M. PAG-Lite also delivers competitive search performance and is the runner-up on DBpedia1536, DBpedia3072, and MajorTOM.
+- *D2. Indexing Time.* Fig. [4](#page-8-1) (left) shows the indexing time. Under the same or larger efC, PAG-Base requires only 20–40% of the indexing time of HNSW, and is faster than SymQG in most cases. PAG-Lite achieves an indexing time comparable to quantization-based methods, which is further reduced
+{8}------------------------------------------------
+<span id="page-8-0"></span>![](_page_8_Figure_1.jpeg)
+<span id="page-8-1"></span>Figure 3. QPS-recall, K = 100. SymQG runs out of memory on MajorTOM. Recall is plotted in logarithmic scale to highlight large values.
+![](_page_8_Figure_3.jpeg)
+Figure 4. Indexing time and peak memory usage. SymQG runs out of memory on MajorTOM.
+to $0.5 \times$ on high-dimensional datasets, where it attains the lowest indexing time.
+D3. Memory Footprint. Fig. 4 shows the memory usage in indexing (middle) and searching (right) phases. PAG-Base uses more memory than HNSW on lower-dimensional datasets, but the difference is negligible on higher-dimensional datasets. PAG-Base consistently uses much less memory than SymQG. PAG-Lite achieves the smallest memory footprint in 4 out of 8 cases for both indexing and searching.
+D4. High-dimensional Scalability. The competitiveness of PAG-Base and PAG-Lite over $d \in [96,3072]$ , as shown in Figs. 3 and 4, is consistent, showcasing its insensitivity to dimensionality. In addition, the advantage of PAG in QPS-recall becomes more pronounced on high-dimensional datasets. In contrast, SymQG reports very low recall on high-dimensional datasets DBpedia1536, DBpedia3072, and DataCompDr, where $d \in \{1536,3072\}$ .
+$\underline{D5.\ Retrieval\ Size\ Robustness}$ . Besides the QPS-recall for K=100 in Fig. 3, we report the results for K=10 in Fig. 16 and K=1000 in Fig. 18. PAG methods are highly
+competitive across the three K values. When K=10, PAG-Base achieves comparable performance and even outperforms SymQG in the high-recall ( $\geq 95\%$ ) region on all datasets except DEEP100M. When K=1000, PAG-Base remains the best, while the performance of SymQG degrades significantly. To highlight the performance comparison, we also plot in Fig. 7 the QPS-recall by varying K from 200 to 800. It can be seen that the gap between PAG-Base and SymQG grows when K moves towards larger values. This observation showcases the robustness of PAG in various applications that differ in retrieval size.
+D6. Online Insertion Support. To evaluate the query processing performance with online insertions, we consider the following workload. We randomly sample from the corpus 10,000 vectors as insertion queries and another 10,000 vectors as search queries. The 20,000 vectors are divided into 20 batches, each with 1,000 vectors. The insertion batches and the search batches are interleaved as a workload, with insertion as the first batch. The rest of the corpus is used to build the initial index. Note that DataCompDr is not OOD in this setting because its original query set (in Table 2) is
+{9}------------------------------------------------
+![](_page_9_Figure_1.jpeg)
+Figure 6. QPS-recall, K = 1000.
+not used. To make the processing times of insertion queries and search queries on the same scale, we set efS = efC and tune these two parameters to control the recall. Fig. 8 compares PAG-Base and HNSW, plotting their QPS-recall performances of insertion and search. Insertion queries are slightly slower to process than search queries for both methods. PAG-Base are much faster than HNSW in both insertion and search speeds, and PAG-Base's insertion is even faster than HNSW's search. Similar to PAG-Base's advantage in search, its speedup over HNSW in insertion can be up to 5 times, demonstrating its efficiency in processing online insertions.
+Ablation Study. We choose four datasets, DBpedia3072, WoltFood, AmazonBooks, and DataCompDr, for ablation study. Among the six modern datasets, they cover the lowest and highest dimensionality, text and images, as well as OOD. Figs. 9 and 10 show the effectiveness of PAG's components. We observe that TFB consistently reduces indexing time and improves search performance. PES further enhances search performance with negligible additional indexing time. The additional memory usage introduced by TFB and PES is very minor and thus not shown here, because TFB does not affect
+the index size and PES barely increases the number of edges.
+## 6. Conclusion
+In this paper, we motivated six critical demands of modern AI applications for ANNS, covering search performance, indexing speed, memory footprint, scalability to dimensionality, robustness against retrieval size, and support of online insertion. To meet these demands, we proposed PAG as a new framework for ANNS. PAG reduces unnecessary distance computations by employing the comparison of exact and approximate distances. The components of PAG are derived from a unified statistical relationship, making its mechanism theoretically explainable. Experiments on modern datasets showcased the superiority of PAG over widely-used ANNS methods as well as state-of-the art solutions, and confirmed the effectiveness of PAG's components.
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+<span id="page-13-4"></span>Zhang, S., Wang, J., Zhou, R., Liao, J., Feng, Y., Zhang, W., Wen, Y., Li, Z., Xiong, F., Qi, Y., et al. MemRL: Selfevolving agents via runtime reinforcement learning on episodic memory. *arXiv preprint arXiv:2601.03192*, 2026. URL <
+{14}------------------------------------------------
+## A. Frequently Used Notations
+<span id="page-14-1"></span>Table 3 shows the notations frequently used in this paper. *Table 3.* Frequently used notations.
+| Symbol | Description | | | | |
+|------------------------------------|-------------------------------------------------------------------|--|--|--|--|
+| $\mathcal{D}$ | Dataset of vectors | | | | |
+| d | Dimension of vectors | | | | |
+| n | Dataset size | | | | |
+| $\boldsymbol{q}$ | Query vector | | | | |
+| K | Retrieval size | | | | |
+| $\boldsymbol{v}$ | Node to be inserted to a graph index | | | | |
+| $N_{\rm out}(\boldsymbol{v})$ | Out-neighbors of $v$ | | | | |
+| $N_{\rm in}(\boldsymbol{v})$ | In-neighbors of $\boldsymbol{v}$ | | | | |
+| $\boldsymbol{u}$ | Candidate neighbor of $v$ to be evaluated | | | | |
+| L | Space partition size | | | | |
+| $\{\boldsymbol{w_i}\}_{i=1}^t$ | $t$ neighbors of $\boldsymbol{u}$ | | | | |
+| $\{\boldsymbol{e_i}\}_{i=1}^t$ | Edges between $\boldsymbol{u}$ and $\{\boldsymbol{w_i}\}_{i=1}^t$ | | | | |
+| $\tau_i$ | Threshold w.r.t. $w_i$ in PRT | | | | |
+| $\delta_i$ | Threshold w.r.t. $w_i$ in PES | | | | |
+| $R_L$ | Result list | | | | |
+| W | Working set | | | | |
+| $R_F, R_T$ | Dual rings of false positives and ejected nodes | | | | |
+| m | Number of projection vectors in each subspace | | | | |
+| ${\mathcal F}$ | Set of concatenated projection vectors | | | | |
+| $\{\boldsymbol{r_i}\}_{i=1}^{m^L}$ | Individual projection vectors | | | | |
+| $r_i^* \in \mathcal{F}$ | Vector having the smallest angle with $w_i - u$ | | | | |
+| $\cos \alpha_i$ | Cosine of the angle between $w_i - u$ and $v - u$ | | | | |
+| $\cos \beta_i$ | Cosine of the angle between $w_i - u$ and $r_i^*$ | | | | |
+| $\cos \theta_i$ | Cosine of the angle between $v-u$ and $r_i^*$ | | | | |
+| Cand(v) | All nodes visited during the insertion of $v$ | | | | |
+| $H(\boldsymbol{v})$ | Nodes in $\operatorname{Cand}(\boldsymbol{v})$ rejected by PES | | | | |
+#### <span id="page-14-0"></span>B. Proof of Theorem 3.1
+In the proof, with slight abuse of notation, we treat u as the origin, and directly use v and each $w_i$ to denote v-u and $w_i-u$ , respectively.
+As such, we have $w_1, \ldots, w_t, v \in \mathbb{S}^{d-1}$ , and $\langle w_i, v \rangle = \cos \alpha_i, i = 1, \ldots, t$ . Suppose that d is divisible by L. Let $w_i = [w_{i1}, \ldots, w_{iL}]^{\top}$ and $v = [v_1, \ldots, v_L]^{\top}$ be the equal-dimension partition of $w_i$ and v, respectively. Suppose that in the l-th subspace, we generate m random projection vectors $\{r_{jl}\}_{j=1}^m$ with the norm $1/\sqrt{L}$ . We use $r_{il}^*$ to denote the nearest projection vector to $w_{il}$ , i.e.,
+$$r_{il}^* = \underset{1 \le j \le m}{\arg\max} \langle r_{jl}, w_{il} \rangle. \tag{7}$$
+Then, we have $\bm{r_i^*} = [\bm{r_{i1}^*}, \dots, \bm{r_{iL}^*}]^\top \in \mathbb{S}^{d-1}.$ We introduce
+$$C_{il} := \frac{\langle \boldsymbol{w_{il}}, \boldsymbol{r_{il}^*} \rangle \sqrt{L}}{\|\boldsymbol{w_{il}}\|} \in [-1, 1]. \tag{8}$$
+Let $\mathbb{E}[C_{il}] = \mu_m$ and $\mathrm{Var}(C_{il}) = \sigma_m^2$ , where $\mu_m$ and $\sigma_m^2$ depend only on m. Note that they implicitly depend on the subspace dimension d/L. When m grows at a sufficiently high rate compared to d/L, $\mu_m \to 1$ and $\sigma_m \to 0$ . Moreover,
+we have
+$$Cov(C_{il}, C_{jl}) = \rho_m(\phi_{ijl})\sigma_m^2 \tag{9}$$
+where $\phi_{ijl}$ denotes the angle between $w_{il}$ and $w_{jl}$ , and $\rho_m := [0,\pi] \to [-1,1]$ denotes the correlation coefficient function depending on $\phi_{ijl}$ . Then, we have
+$$Y_i := \Sigma_{l=1}^L \langle \boldsymbol{r_{il}^*}, \boldsymbol{w_{il}} \rangle = \Sigma_{l=1}^L \frac{\|\boldsymbol{w_{il}}\|}{\sqrt{L}} C_{il}.$$
+(10)
+By the assumption that $\|\boldsymbol{w_{il}}\| = (1 + o(1))/\sqrt{L}$ , we have
+$$\mathbb{E}[Y_i] = \mu_m(1 + o(1)). \tag{11}$$
+$$Var(Y_i) = \frac{\sigma_m^2}{L}(1 + o(1)).$$
+(12)
+<span id="page-14-2"></span>
+$$Cov(Y_i, Y_j) = \frac{\sigma_m^2}{L^2} \sum_{l=1}^{L} \rho_m(\phi_{ijl}) (1 + o(1)).$$
+(13)
+Then, $v_l$ can be decomposed as $v_l = v_{il}^{\parallel} + v_{il}^{\perp}$ , where $v_{il}^{\parallel}$ and $v_{il}^{\perp}$ are defined as follows.
+$$v_{il}^{\parallel} = \frac{\langle v_l, w_{il} \rangle}{\|w_{il}\|^2} w_{il}, \quad v_{il}^{\perp} \perp w_{il}.$$
+(14)
+Then, we can define $Z_i$ as follows.
+$$Z_i := \langle r_i^*, v \rangle = Z_i^{(1)} + Z_i^{(2)}$$
+(15)
+where $Z_i^{(1)}$ and $Z_i^{(2)}$ are defined as follows.
+$$Z_i^{(1)} := \sum_{l=1}^{L} \frac{\langle \boldsymbol{v_l}, \boldsymbol{w_{il}} \rangle}{\|\boldsymbol{w_{il}}\| \sqrt{L}} C_{il}.$$
+(16)
+$$Z_i^{(2)} := \sum_{l=1}^L \langle r_{il}^*, v_{il}^{\perp} \rangle. \tag{17}$$
+$$\mathbb{E}[Z_i^{(1)}] = \frac{\mu_m}{\sqrt{L}} \sum_{l=1}^L \frac{\langle \boldsymbol{v}_l, \boldsymbol{w}_{il} \rangle}{\|\boldsymbol{w}_{il}\|}$$
+$$= \mu_m \cos \alpha_i + o(1).$$
+(18)
+By symmetry, we have $\mathbb{E}[Z_i^{(2)}|\{C_{il}\}_{l=1}^L]=0$ . Then, by the independence of subspaces, we have
+$$\mathbb{E}[Z_i] = \mu_m \cos(\alpha_i) + o(1). \tag{19}$$
+$$Var(Z_i^{(1)}) = \frac{\sigma_m^2}{L} \sum_{l=1}^{L} \frac{\langle v_l, w_{il} \rangle^2}{\|w_{il}\|^2}.$$
+(20)
+We take the orthogonal decomposition of $r_{il}^*$ as $r_{il}^* = r_{il}^{\parallel} + r_{il}^{\perp}$ , where $r_{il}^{\parallel}$ has the same direction with $w_{il}$ . Let $r_{il}^{\perp} = \|r_{il}^{\perp}\|\zeta$ , where $\|\zeta\| = 1$ is a random vector in a (d/L - 1)-dimensional subspace. We have
+$$\mathbb{E}_{\zeta}[\langle \zeta, v_{il}^{\perp} \rangle^{2}] = \frac{\|v_{il}^{\perp}\|^{2} L}{d - L}.$$
+(21)
+{15}------------------------------------------------
+By symmetry, $\mathbb{E}[\langle r_{il}^{\perp}, v_{il}^{\perp} \rangle \mid C_{il}] = 0$ . Then, we have
+$$\operatorname{Var}(\langle \boldsymbol{r}_{il}^*, \boldsymbol{v}_{il}^{\perp} \rangle) = \mathbb{E}_{C_{il}} \left[ \mathbb{E}[\langle \boldsymbol{r}_{il}^{\perp}, \boldsymbol{v}_{il}^{\perp} \rangle^2 \mid C_{il}] \right]. \tag{22}$$
+Using $\mathbb{E}[\|\boldsymbol{r}_{il}^{\perp}\|^2] = \frac{1}{L}(1 - \mathbb{E}[C_{il}^2])$ , we have
+<span id="page-15-1"></span>
+$$Var(Z_i^{(2)}) = \frac{1 - (\sigma_m^2 + \mu_m^2)}{d - L} \sum_{l=1}^{L} \| v_{il}^{\perp} \|^2.$$
+(23)
+Because $Cov(Z_i^{(1)}, Z_i^{(2)}) = 0$ , we have
+$$Var(Z_i) = Var(Z_i^{(1)}) + Var(Z_i^{(2)}).$$
+(24)
+We define $\epsilon_m:=\sqrt{1-\mathbb{E}[C_{il}^2]}$ . From Eq. (23), we can see that $\mathrm{Var}(Z_i^{(2)})=O(\epsilon_m^2/L)$ .
+We now analyze the covariance structure. We cannot assume $Z^{(1)}$ and $Z^{(2)}$ are uncorrelated across different indices $i \neq j$ . Instead, we decompose the covariance matrix as
+<span id="page-15-3"></span>
+$$Cov(Z_i, Z_j) = Cov(Z_i^{(1)}, Z_j^{(1)}) + R_{ij}$$
+(25)
+where the residual term $R_{ij}$ contains the noise auto-covariance and cross-terms:
+$$R_{ij} = \text{Cov}(Z_i^{(2)}, Z_j^{(2)}) + \text{Cov}(Z_i^{(1)}, Z_j^{(2)}) + \text{Cov}(Z_i^{(2)}, Z_j^{(1)}).$$
+(26)
+Because ${\rm Var}(Z^{(1)})=O(1/L)$ and ${\rm Var}(Z^{(2)})=O(\epsilon_m^2/L)$ , by the Cauchy-Schwarz inequality, we have
+$$|\text{Cov}(Z_i^{(1)}, Z_j^{(2)})| \le \sqrt{\text{Var}(Z_i^{(1)})\text{Var}(Z_j^{(2)})} = O\left(\frac{\epsilon_m}{L}\right).$$
+Thus, the entire residual term satisfies $R_{ij} = O(\epsilon_m/L)$ .
+Now, let us turn to $\mathrm{Cov}(Y_i,Z_j)$ . When i=j, by conditional expectation, $\mathrm{Cov}(Y_i,Z_j^{(2)})=0$ . When $i\neq j$ , if $m\to\infty$ , $\sigma_m^2+\mu_m^2\to 1$ and $\mathrm{Var}(Z_i^{(2)})\to 0$ . Thus, $\mathrm{Cov}(Y_i,Z_j^{(2)})\to 0$ . Then, as $m\to\infty$ , we have
+<span id="page-15-2"></span>
+$$Cov(Y_i, Z_j) \to \frac{\sigma_m^2}{L} \sum_{l=1}^{L} \rho_m(\phi_{ijl}) \langle \boldsymbol{v_l}, \boldsymbol{w_{jl}} \rangle (1 + o(1)).$$
+(28)
+Based on the above analysis, we consider
+$$\boldsymbol{\xi_{l}} = \left[ \frac{\|\boldsymbol{w_{1l}}\|}{\sqrt{L}} C_{1l}, \dots, \frac{\|\boldsymbol{w_{tl}}\|}{\sqrt{L}} C_{tl}, \langle \boldsymbol{r_{1l}^*}, \boldsymbol{v_l} \rangle, \dots, \langle \boldsymbol{r_{tl}^*}, \boldsymbol{v_l} \rangle \right].$$
+(29)
+By the assumptions that $\|v_{il}\|$ and $\|w_{il}\|$ equal $(1 + o(1))/\sqrt{L}$ , it can be seen that $\xi_l$ satisfies the Lyapunov condition. By the independence of difference subspaces, we use the Lindeberg-Feller CLT and obtain the following result.
+$$\sqrt{L}([Y_1, \dots, Y_t, Z_1, \dots, Z_t]^{\top} - \bar{\boldsymbol{\mu}}) \xrightarrow[L \to \infty]{d} \mathcal{N}(0, \bar{\Sigma}_{m,L})$$
+(30)
+where $\bar{\mu} = [\mu_m, \dots, \mu_m, \mu_m \cos \alpha_1, \dots, \mu_m \cos \alpha_t]^\top$ , and $\bar{\Sigma}$ is represented by block sub-matrices
+$$\bar{\Sigma} = \begin{pmatrix} \Sigma_{YY} & \Sigma_{YZ} \\ \Sigma_{ZY} & \Sigma_{ZZ} \end{pmatrix} \tag{31}$$
+where the elements of $\Sigma_{YY}$ , $\Sigma_{YZ}$ and $\Sigma_{ZZ}$ are L times the covariance computed in Eqs. (13), (28), and (25), respectively.
+Based on this result, Z | Y is still a multivariate normal distribution. We have
+$$\mathbb{E}[\mathbf{Z}|\mathbf{Y}] = \boldsymbol{\mu}_Z + \boldsymbol{\Sigma}_{ZY} \boldsymbol{\Sigma}_{YY}^{-1} (\mathbf{Y} - \boldsymbol{\mu}_Y). \tag{32}$$
+$$Cov(\mathbf{Z}|\mathbf{Y}) = (\Sigma_{ZZ} - \Sigma_{ZY}\Sigma_{YY}^{-1}\Sigma_{YZ})/L.$$
+(33)
+Let D be the diagonal matrix defined as follows
+$$\mathbf{D} = \operatorname{diag}[\cos \alpha_1, \dots, \cos \alpha_t]. \tag{34}$$
+Because $\Sigma_{ZY} \to \mathbf{D}\Sigma_{YY}$ , when $\mathbf{Y} = [\cos \beta_1, \dots, \cos \beta_t]^\top$ , we have
+$$\mathbb{E}[\mathbf{Z}|\mathbf{Y}] \to [\cos \alpha_1 \cos \beta_1, \dots, \cos \alpha_t \cos \beta_t]^\top.$$
+(35)
+$$Cov(\mathbf{Z}|\mathbf{Y}) \to (\mathbf{\Sigma}_{\mathbf{Z}\mathbf{Z}} - \mathbf{D}\mathbf{\Sigma}_{\mathbf{Y}\mathbf{Y}}\mathbf{D})/L = \mathbf{R}/L$$
+(36)
+where $\mathbf{R}$ represents the scaled residual matrix with entries $L \cdot R_{ij} = O(\epsilon_m)$ . Then, we conclude.
+**Remarks.** For each fixed i, if we apply a random rotation matrix to $w_i - u$ and v - u in $\mathbb{R}^d$ , by the spherical concentration inequality, we can derive the following inequalities.
+$$P\left(\left|\|(\boldsymbol{w_i} - \boldsymbol{u})_l\|^2 - \frac{1}{L}\right| \ge \tilde{\epsilon}\right) \le 2e^{-c_1 d\tilde{\epsilon}^2}$$
+(37)
+$$P\left(\left|\langle (\boldsymbol{v}-\boldsymbol{u})_l, (\boldsymbol{w_i}-\boldsymbol{u})_l \rangle - \frac{\cos \alpha_i}{L}\right| \ge \tilde{\epsilon}\right) \le 2e^{-c_2 d\tilde{\epsilon}^2}$$
+(38)
+where $c_1$ and $c_2$ are constants. As $L \to \infty$ and $d/(L^2) \to \infty$ , we have the following results:
+$$\|(\boldsymbol{w_i} - \boldsymbol{u})_l\| = (1 + o_p(1))/\sqrt{L}.$$
+(39)
+$$\langle (\boldsymbol{v} - \boldsymbol{u})_l, (\boldsymbol{w_i} - \boldsymbol{u})_l \rangle = \frac{\cos \alpha}{L} + o_p(\frac{1}{L}).$$
+(40)
+This implies that $L=\sqrt{d}$ is a balanced choice, which is consistent with the setting in our experiments.
+#### <span id="page-15-0"></span>C. Related Work
+We first supplement Sec. 1.1 with more discussions, and then introduce the preliminaries on probabilistic routing.
+{16}------------------------------------------------
+#### C.1. Discussions on ANNS Solvers
+911912
+#### C.1.1. VECTOR QUANTIZATION
+Quantization-based methods for ANNS have a long history. Representative approaches can be found in (Jégou et al., 2011; Ge et al., 2014; Babenko & Lempitsky, 2016; Guo et al., 2020a; Gao & Long, 2024; Gao et al., 2025). Here, we briefly introduce the basic idea of the most widely-used method, Product Quantization (PQ) (Jégou et al., 2011).
+Let $\mathcal{D} \subset \mathbb{R}^d$ be a d-dimensional dataset and let $q \in \mathbb{R}^d$ be a query vector. PQ divides each data vector $x \in \mathcal{D}$ into L sub-vectors, i.e., $x = (x_1, x_2, \ldots, x_L)$ , where each sub-vector $x_j \in \mathbb{R}^{d'}$ has dimension d' = d/L. In this way, PQ constructs L subspaces of dimension d', each containing the corresponding sub-vectors of all data points.
+In each subspace, every sub-vector is assigned to one of $2^k$ quantized sub-vectors. Equivalently, each sub-vector is represented by a sub-codeword of length k bits. The set of all sub-codewords in the j-th subspace is referred to as a codebook and is denoted by $\mathcal{C}_j$ . By concatenating the sub-codewords across all subspaces, each original vector is represented by a codeword of length kL, taking values in the Cartesian product $\mathcal{C}_1 \times \cdots \times \mathcal{C}_L$ .
+During the query phase, PQ first computes the distances between the query $\boldsymbol{q}$ and all quantized sub-vectors in each subspace. For each data vector $\boldsymbol{x}$ , PQ then sums the corresponding L subspace distances to obtain an approximate distance between $\boldsymbol{x}$ and $\boldsymbol{q}$ . After computing the approximate distances for all data vectors, PQ ranks them accordingly and returns the most promising candidates.
+### C.1.2. SIMILARITY GRAPH
+The workflow of graph-based methods is roughly as follows. In the indexing phase, a similarity graph is constructed as the index structure, where data points serve as nodes and edges connect pairs of nearby nodes. A search can move directly from one node to another only if an edge exists between them. In the query phase, the node with the highest priority in a priority queue is selected, and all of its connected neighbors are visited. During this process, the priority queue is updated whenever a node closer to the query is discovered. The search ends when all nodes in the priority queue have been visited. Finally, the top-K points in the priority queue are returned.
+Among various graph-based methods, HNSW (Malkov & Yashunin, 2020) is a notable one that employs a multi-layer hierarchical structure to achieve rapid routing. NSG (Fu et al., 2019) optimizes the graph topology to ensure the existence of monotonic paths towards a central entry point, thereby enhancing search efficiency. Vamana, which was introduced along with DiskANN (Subramanya et al., 2019), iteratively refines a random graph into a high-performance graph struc-
+ture. Despite their structural differences, these methods converge on a similar edge selection criterion, namely **RobustPrune**, which prioritizes directional diversity over simple proximity to ensure efficient navigation through high-dimensional space.
+In addition to the three graphs mentioned above, many graph-based methods have been proposed recently (Lu et al., 2021; Gao & Long, 2023; Xie et al., 2025; Wang et al., 2025b). Despite different designs, most of them rely on existing graphs, such as HNSW.
+#### C.1.3. QUANTIZED GRAPH
+As stated in the main body of this paper, QG can achieve high performance on some datasets for small values of K, and in certain cases can be $4 \times -10 \times$ faster than HNSW. However, such improvement is often not robust and suffers from several limitations: (1) Sensitivity to data distribution. The effectiveness of QG relies on the assumption that the ranking induced by quantized distances is sufficiently close to the ranking under the original distances. Unfortunately, this assumption often does not hold, especially for many modern real-world datasets where semantic embeddings are increasingly complex. (2) **Sensitivity to** K. As K increases, the performance of QG degrades significantly. When K reaches the thousand scale, QG generally has no significant advantage over HNSW. (3) Very large space cost. To improve the accuracy of QG, more bits are required to represent the quantized vectors. As a result, the memory consumption of QG is typically at least $2 \times$ larger than HNSW, compromising the use of QG for large datasets.
+#### C.2. Preliminaries of Routing Test
+#### C.2.1. RANDOM PROJECTION FOR ANGLE ESTIMATION
+Because the norms can be pre-computed, estimating the $\ell_2$ distance is equivalent to estimating the cosine of the angle between vectors. In high-dimensional Euclidean spaces, angle estimation via random-projection techniques has been extensively studied, with Locality Sensitive Hashing (LSH) (Indyk & Motwani, 1998; Andoni & Indyk, 2005; 2008) being one of the most influential approaches. Among various LSH methods, SimHash (Charikar, 2002) is a representative one. Its core idea is to generate multiple random hyperplanes that partition the space into cells, so that vectors falling into the same cell are likely to form a small angle with each other. Subsequent studies have proposed more refined strategies for angular distance estimation. In particular, Andoni et al. (2015) introduced Falconn, an LSH method that identifies the projection vector yielding the largest or smallest projection value for a given data vector and uses the corresponding projection index as the hash value. This design leads to substantially improved search performance compared to SimHash. Building on this idea, Pham (2021) further incorporated Con-
+{17}------------------------------------------------
+<span id="page-17-1"></span>![](_page_17_Figure_1.jpeg)
+938939
+944945946
+949950
+Figure 11. DataCompDr data (sampled 50,000 vectors) and query (1,000 vectors) distributions.
+comitants of Extreme Order Statistics (CEOs) to explicitly identify the projection achieving the maximum or minimum inner product with the data vector, and to record the associated extreme projected value. By exploiting this additional information beyond a discrete hash value, a more accurate estimation of angular distance can be achieved (Pham & Liu, 2022).
+#### C.2.2. ROUTING TEST IN SIMILARITY GRAPHS
+Due to its simplicity and ease of implementation, CEOs has been adopted in a variety of similarity search tasks (Pham, 2021; Andoni et al., 2015; Xu & Pham, 2024). Beyond standalone similarity estimation, CEOs has also been leveraged to accelerate similarity graphs, which constitute one of the most effective structures for ANNS. By swapping the roles of the query and data vectors in the original CEOs formulation, Lu et al. (2024) developed a space-partitioning technique and proposed the PEOs test. This test enables probabilistic comparisons between the objective angle and a fixed threshold, and has been integrated into the routing procedures of similarity graphs. Specifically, for every visited node u, we send all the out-neighbors of u to PEOs. Only the exact distances between q and the nodes that pass the PEOs test are computed. In the experiments of (Lu et al., 2024), only around 25% nodes can pass the PEOs test. As a result, substantial improvements in search performance brought by PEOs were reported over similarity graphs such as HNSW (Malkov & Yashunin, 2020) and NSSG (Fu et al., 2022). More recently, Lu et al. (2025) proposed a new test function, KS2, which achieves higher test accuracy while maintaining a shorter test time compared with PEOs. KS2 employs a projection structure similar to that of PEOs, but additionally incorporates the reference angle into the testing procedure. Lu et al. (2025) further showed that, without introducing additional assumptions, the test guarantees a success probability of at least 0.5 when deciding whether the exact distance between the objective node and q should be evaluated.
+*Table 4.* PAG parameter settings.
+<span id="page-17-2"></span>
+| Dataset | PAG-Base $(efC, M, L)$ | PAG-Lite $(efC, M, L)$ | | | | | |
+|----------------------------|-----------------------------------|---------------------------------|--|--|--|--|--|
+| Modern Datasets | | | | | | | |
+| DBpedia1536<br>DBpedia3072 | (1000, 32, 128)<br>(1000, 32, 64) | (100, 32, 128)<br>(100, 32, 64) | | | | | |
+| WoltFood | (2000, 128, 64) | (100, 32, 04)<br>(100, 64, 32) | | | | | |
+| DataCompDr<br>AmazonBooks | (1000, 32, 64)<br>(2000, 64, 64) | (100, 32, 64)<br>(100, 64, 32) | | | | | |
+| MajorTOM | (1000, 32, 64) | (100, 04, 32) $(100, 16, 64)$ | | | | | |
+| Legacy Datasets | | | | | | | |
+| Word2Vec | (8000, 64, 32) | (200, 64, 32) | | | | | |
+| GIST | (1000, 64, 96) | (64, 32, 96) | | | | | |
+| GloVe | (2000, 64, 32) | (100, 32, 32) | | | | | |
+| ImageNet | (2000, 64, 16) | (200, 16, 16) | | | | | |
+| SIFT10M | (1000, 32, 16) | (100, 16, 16) | | | | | |
+| DEEP100M | (1000, 32, 16) | (100, 16, 16) | | | | | |
+### <span id="page-17-0"></span>**D. Experimental Setup Details**
+#### **D.1. Environment**
+All experiments were conducted on a machine equipped with an Intel Xeon Platinum 8276L CPU, which supports AVX-512 instructions and provides 112 hardware threads. The system was configured with 754 GB of DDR4 ECC memory and runs Ubuntu 22.04. For indexing, all 112 threads were used, while search was performed using a single CPU thread, in line with the standard setting in ANN-Benchmarks (Bernhardsson et al., 2015).
+#### D.2. Baselines
+#### D.2.1. SELECTION OF BASELINES
+For quantization-based methods, we choose IVF-PQFS (Jégou et al., 2011), ScaNN (Guo et al., 2020a), and RabitQ+ (Gao et al., 2025) as baselines, where RabitQ+ (Gao et al., 2025) is an improved version of RabitQ (Gao & Long, 2024). For graph-based methods, we choose HNSW (Malkov & Yashunin, 2020), Vamana (Subramanya et al., 2019), and SymphonyQG (Gou et al., 2025), where SymphonyQG has shown its superiority over LVQ (Aguerrebere et al., 2023) and NGT-QG (Yahoo! Japan, 2023). On the other hand, HNSW+KS2 (Lu et al., 2025) has been shown to perform better than HNSW+PEOs (Lu et al., 2024), and KS2 can be approximately viewed as our PRT without TFB, modulo threshold-setting differences. Thus, for a fair comparison, we re-implement the KS2 component within our framework and report results for PAG versus PRT only in our ablation study.
+### D.2.2. PARAMETER SETTING
+For graph-based methods, we used large graph construction parameters to ensure their best search performance. To ensure a fair comparison of indexing time, the efC parameter in PAG-Base was set to a similar or even larger value than HNSW.
+{18}------------------------------------------------
+<span id="page-18-0"></span>![](_page_18_Figure_1.jpeg)
+<span id="page-18-1"></span>Figure 12. QPS-recall of PAG-Base under varying space partition size L, K = 100. M and efC values are given in Table 4, PAG-Base.
+![](_page_18_Figure_3.jpeg)
+Figure 13. Indexing time and peak memory usage of PAG-Base under varying space partition size $L.\ M$ and efC values are given in Table 4, PAG-Base.
+<span id="page-18-2"></span>![](_page_18_Figure_5.jpeg)
+Figure 14. QPS-recall of PAG-Base under varying (M, efC), K = 100. L values are given in Table 4, PAG-Base.
+![](_page_18_Figure_7.jpeg)
+Figure 15. Indexing time and peak memory usage of PAG-Base under varying (M, efC). L values are given in Table 4, PAG-Base.
+(1) **HNSW**: efC=1024. M=32. The only exception is MajorTOM, where efC=512 due to the long indexing time on this dataset.
+<span id="page-18-3"></span>
+- (2) **Vamana**: By default, R=64. L=1024. $\alpha=1.2$ . For better QPS-recall performance on MajorTOM and DEEP100M, we choose R=100 and L=250 on MajorTOM, and R=110 and L=200 on DEEP100M.
+- (3) **SymQG**: efC = 1024. There are two options for the degree, 32 and 64, resulting in two baselines, SymQG (32)
+- and SymQG (64).
+- (4) **ScaNN**: We adopt the recommended settings from its GitHub repository (Guo et al., 2020b).
+- (5) **RabitQ+**: Following the same setting as in (Gao et al., 2025), $b=8.\ k=16,384$ for DataCompDr and 4096 for the other datasets.
+- (6) **IVFPQFS**: We test combinations of $nlist \in \{1024, 4096, 16384\}, k_-factor \in \{64, 128, 256\}, and$
+{19}------------------------------------------------
+M candidates ∈ [48, 384]. A combination is chosen for good QPS-recall performance on each dataset.
+(7) PAG: PAG has two versions, PAG-Base and PAG-Lite. Their parameter settings are listed in Table [4.](#page-17-2) Users can adjust the values of efC, M, and L by taking into account the indexing time, memory footprint, and search efficiency.
+other three datasets. PAG-Base also uses less memory than SymQG. The QPS-recall performance of PAG-Lite is generally better than HNSW and Vamana. Meanwhile, PAG-Lite achieves the smallest indexing time, and is competitive in memory footprint.
+## <span id="page-19-0"></span>E. Additional Experimental Results
+## E.1. Effect of Space Partition Size L
+From the QPS-recall results in Fig. [12,](#page-18-0) we can see that, within a moderate range, increasing L leads to better search performance, at the cost of a higher memory footprint. As shown in Fig. [13](#page-18-1) (left), larger L values also results in longer indexing time, due to the faster search speed for each inserted vector. On the other hand, this causes a moderately larger index size, which is reflected in the memory footprint reported in Fig. [13](#page-18-1) (middle and right). As analyzed in Appendix [B,](#page-14-0) L = √ d is an appropriate choice.
+### E.2. Effect of M and efC
+We vary parameters M and efC and plot the QPS-recall in Fig. [14.](#page-18-2) Larger M and efC value result in faster query processing speed, and the advantage is more remarkable when users require high recall values. As shown in Fig. [15,](#page-18-3) increasing M and efC generally leads to more indexing time as well as larger memory footprint. This is expected, because they control the out-degree and the number of nodes visited for each insertion. The only exception is DataCompDr, where a largerM does not necessarily mean a slower indexing speed. This is because the search speed for each inserted vector is accelerated despite a larger out-degree.
+### <span id="page-19-1"></span>E.3. Evaluation on Additional Legacy Datasets
+We report the results on four additional legacy datasets, Word2Vec, ImageNet, GIST, and SIFT10M, which are commonly used for ANNS evaluation. Figs. [16,](#page-20-0) [17,](#page-20-2) and [18](#page-20-1) show the QPS-recall performances when K = 10, 100, and 1000. Fig. [19](#page-20-3) shows the indexing time and memory footprint. From the results in figures, we have the following observations. PAG-Base maintains its competitiveness in QPS-recall performance, as we have witnessed in the experiments on other datasets. On Word2Vec and ImageNet, where graph-based methods and SymQG struggle to achieve high recalls, PAG-Base has a significant advantage in QPS. On GIST and SIFT10M, SymQG performs better than PAG-Base in QPS when K = 10 and K = 100, because the two datasets are sparse and well-suited for vector quantization. When K = 1000, PAG-Base outperforms SymQG by a large margin. For indexing speed, PAG-Base is slower than graphbased methods and SymQG on Word2Vec but is faster on the
+{20}------------------------------------------------
+<span id="page-20-3"></span>
+<span id="page-20-0"></span>![](_page_20_Figure_1.jpeg)
+Figure 16. QPS-recall on additional legacy datasets, K = 10.
+<span id="page-20-2"></span>![](_page_20_Figure_3.jpeg)
+Figure 17. QPS-recall on additional legacy datasets, K = 100.
+<span id="page-20-1"></span>![](_page_20_Figure_5.jpeg)
+Figure 18. QPS-recall on additional legacy datasets, K = 1000.
+![](_page_20_Figure_7.jpeg)
+Figure 19. Indexing time and peak memory usage on additional legacy datasets.