{"id":"70351d56-1694-474a-81b3-c742b5078371","arxiv_id":"2502.06689","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"Neumann eigenmaps embed a landmark subgraph with reflecting-random-walk eigenvectors, but the central distance-preservation theorem is not supported by the proof.","lead":"Neumann eigenmaps offer a new way to embed a subset of data points by treating the remaining points as a reflective boundary for a random walk. The paper claims this embedding preserves a diffusion distance, but the proof of that claim contains a mathematical error.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem III.3 fails already at t=0: the left eigenvectors y_l used in Definition III.3 are not orthonormal under the T^{-1} metric, so the asserted isometry does not hold on non-regular graphs.","rationale":"The paper's contribution is the claimed isometric landmark embedding, so the truth of Theorem III.3 is load-bearing. The reader's REJECT verdict is correct. My independent check at t=0 shows the stated theorem is false in a simple, explicitly computable case, and the source is exactly the use of left eigenvectors with the stated metric and the discarding of a non-constant first eigenvector. The numerical experiments may still be useful heuristically, but they do not repair the central theoretical claim. I agree with the reader's weakest_assumption, and I add a sharper algebraic diagnosis: the biorthogonality Y^JV=I does not imply V^J T^{-1} V=I, and the appendix's proof relies on that false implication. The correct fix would be to use the right eigenvectors with a row-wise diffusion distance, or equivalently to redefine the metric and weights so that the left eigenvectors are orthonormal; either way, the current statement of Theorem III.3 needs revision. Since the reader already recommended REJECT with moderate confidence, my stress-test leaves that verdict unchanged.","tokens_in":10027,"tokens_out":16249,"duration_ms":138258,"concrete_test":"Let S be the 3-vertex path with unit edge weights and no boundary (B=0), so R is the standard random walk matrix and d=(1,2,1). For t=0, Theorem III.3 with i=1, k=2 gives LHS = 1/d_1 + 1/d_2 = 1.5. Compute the Neumann eigenvectors: u_1=(1,sqrt(2),1)/2, u_2=(1,0,-1)/sqrt(2), u_3=(1,-sqrt(2),1)/2, hence y_l=D^{1/2}u_l. Definition III.3 with all retained vectors gives RHS = (y_2(1)-y_2(2))^2 + (y_3(1)-y_3(2))^2 = (1/sqrt(2))^2 + (3/2)^2 = 2.75. Since 1.5 != 2.75, equation (III.4) is false. Repeating at t=1 gives LHS 2.125 versus RHS 2.25, confirming the failure is not special to t=0.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is the isometry (III.4). For t=0, R^0=I, so the left side of (III.4) is 1/d_i+1/d_k. According to Definition III.3, the right side is sum_{l=2}^{|S|}(y_l(i)-y_l(k))^2, with y_l=T^{1/2}u_l and u_l orthonormal eigenvectors of N. Since YY^J=T, the full sum including l=1 equals d_i+d_k-2(W_S)_{ik}; the discarded first vector y_1 is proportional to the degree vector d, not to the constant vector, so its contribution c(d_i-d_k)^2 is generally nonzero and the claimed identity cannot hold. The Appendix D proof contains the corresponding algebraic error: with V=T^{-1/2}U, one has V^J T^{-1}V = U^J T^{-2}U, not I; the displayed simplification T^{1/2}T^{-1}T^{1/2}=I is applied to the wrong factors. The correct diffusion-map coordinates for a distance weighted by 1/d_j are the right eigenvectors v_l=T^{-1/2}u_l paired with the row-wise transition probabilities, not the left eigenvectors y_l paired with column-wise probabilities as in (III.4). The failure is structural and already visible on a three-vertex path.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes NeuMaps, a landmark-based embedding computed from the eigenvectors of a renormalized Neumann Laplacian on an induced subgraph. The central theoretical claim is Theorem III.3: the Euclidean distance between NeuMap coordinates g_t(x_i) and g_t(x_k) equals the weighted sum of reflecting-random-walk transition probabilities, sum_j (p_{j,i} - p_{j,k})^2 / d_j. The paper also presents numerical experiments on digit classification and molecular dynamics, comparing NeuMaps with Roseland and standard diffusion maps.","tokens_in":10321,"tokens_out":8474,"duration_ms":69591,"significance":"If the isometry of Theorem III.3 were correct, NeuMaps would provide a principled landmark embedding that provably recovers a diffusion distance while retaining the computational advantages of subsampling. The paper also connects the Neumann boundary condition to Nyström extension and includes a code repository. However, the main theoretical result is invalid as stated: the embedding uses left eigenvectors of the transition matrix, and the claimed isometry fails already at t=0 on non-regular graphs. The numerical comparisons are therefore not backed by the asserted geometric guarantee, and the contribution currently lacks a demonstrated theoretical foundation.","major_comments":[{"comment":"Theorem III.3 is false for the embedding defined in Definition III.3. The coordinates are g_t(x_i) built from y_l = T^{1/2}u_l, which are the left eigenvectors of the reflecting random walk matrix R. The first left eigenvector is proportional to the degree vector d, not to the constant vector. Consequently, dropping the first eigenvector removes a nonconstant component, contrary to standard diffusion-map practice. For t=0 the left side of (III.4) is 1/d_i + 1/d_k, while the right side equals d_i + d_k - (y_1(i) - y_1(k))^2, and (y_1(i) - y_1(k))^2 is generally nonzero when d_i ≠ d_k. The identity already fails on a three-vertex path, so the error is structural.","section":"Section III, Definition III.3 and Theorem III.3"},{"comment":"The algebraic proof contains a load-bearing error. With V = T^{-1/2}U and R^t = T^{-1/2}U Σ^t U^T T^{1/2}, the expression (R^t)^T T^{-1} R^t equals T^{1/2} U Σ^t U^T T^{-2} U Σ^t U^T T^{1/2}, not the simplified form obtained in the displayed calculation. The underbrace T^{1/2}T^{-1}T^{1/2}=I is applied to the wrong factors; the actual combination is U^T T^{-2} U, which is not the identity. This invalid cancellation is essential to the claimed isometry.","section":"Appendix D, proof of Theorem III.3"},{"comment":"The numerical experiments are run with the embedding of Definition III.3, which does not satisfy the isometry asserted in Theorem III.3. As a result, the reported NMI, ACC, and standard-error numbers do not validate the theoretical claim. Regenerating the experiments with the corrected embedding (right eigenvectors v_l = T^{-1/2}u_l) is necessary before the experimental claims can be assessed, since the algorithm itself would change.","section":"Section IV, numerical experiments"}],"minor_comments":[{"comment":"The word 'conisder' in the discussion of the subsampling-uniformly-in-time scheme is a typo; it should read 'consider'.","section":"Section IV, paragraph 2, digit classification"},{"comment":"Figures 1 and 2 are referenced in the text but are not included in the provided manuscript version; please ensure they are present in the final submission.","section":"General, figures"},{"comment":"The proof of Theorem III.3 introduces symbols C, B, W, and Ψ without definition; the notation should be aligned with the main text (U, V, Y, Σ) to make the argument readable.","section":"Appendix D, proof notation"},{"comment":"References [2] and [5] appear to refer to the same paper (Chung and Yau, 1997, 'Eigenvalue inequalities for graphs and convex subgraphs'); if so, the duplicate should be removed or the citations updated.","section":"References"},{"comment":"The sentence contrasting the reflecting random walk with 'the one in Roseland' would benefit from an explicit citation to the Roseland algorithm, as the current reference list does not clearly identify it.","section":"Section III, Remark III.1"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is false as stated, but the error appears fixable by replacing the left eigenvectors with right eigenvectors in the definition of NeuMaps. If the authors make that change, the isometry becomes a standard diffusion-map result for the reflecting random walk, and the numerical experiments would need to be redone. The current manuscript, however, cannot be accepted in its present form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline is that the central isometry theorem (Theorem III.3) is not correct as stated. The paper defines NeuMap coordinates from the left eigenvectors y_l = T^{1/2} u_l and claims the Euclidean distance between coordinates equals the column-wise reflecting-random-walk distance weighted by 1/d_j. At t=0 the claim would require sum_{l≥2}(y_l(i)-y_l(k))^2 = 1/d_i + 1/d_k. But the full sum including l=1 is d_i+d_k, and the discarded first term is proportional to (d_i-d_k)^2, which is generally nonzero. The appendix's derivation uses V^T T^{-1} V = I, which is false; only Y^T T^{-1} V = I holds. (The stress-test says the y_l are not orthonormal under T^{-1}; they are — the problem is that the embedding's metric is the unweighted Euclidean one, not T^{-1}.) This is a load-bearing flaw: the advertised rigorous distance guarantee rests entirely on this theorem.\n\nThe paper does have real merit. Combining the Neumann Laplacian (and its reflected random walk) with landmark diffusion maps is a new and sensible idea, and the contrast with Roseland is clearly articulated. The connection between the discrete Neumann boundary condition and Nyström extension (Remark III.2) is a genuinely nice observation. The writing is clean, the related work is handled responsibly, and the authors ship code and data for the numerical experiments. Those experiments, while proof-of-concept rather than exhaustive, do suggest the construction can help with landmark-based embedding and erasure stability.\n\nOther concerns are minor. The experiments are limited to two datasets with no error bars or hyperparameter sensitivity. There is no circular reasoning problem; this is a straightforward but flawed derivation. Self-citations are not an issue.\n\nI think the paper deserves a serious referee, not a desk reject. The conceptual contribution is worth keeping, and the isometry error is specific enough that a competent referee can tell the authors exactly where to fix it — likely by switching to right eigenvectors or by redefining the distance and retaining the first vector. As it stands, though, the central claim fails, and I would not cite the theorem in its current form. For you: worth a look if you work on spectral landmark methods, but keep the t=0 check in hand.","headline":"The isometry theorem that carries the paper's main claim is wrong — the discarded first left eigenvector is not constant and the appendix misuses V^T D^{-1} V = I — but the underlying NeuMap construction and Nyström observation are worth a referee's time.","tokens_in":10803,"tokens_out":6633,"would_cite":false,"duration_ms":52448,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C50","60J10"],"pacs":[],"model":"deepseek-v4-flash","headline":"NeuMaps are a landmark embedding whose Euclidean distances equal the reflecting walk's diffusion distance.","keywords":["Neumann eigenmaps","landmark embedding","diffusion maps","reflecting random walk","Neumann Laplacian","Nyström extension","manifold learning","spectral graph theory"],"falsifier":"Compute both sides of the identity in Theorem III.3 for a small non-regular graph, for instance a path of three landmarks with one boundary vertex attached to the middle landmark. If the squared NeuMap distance does not equal $\\sum_j (p_{ji}^t-p_{jk}^t)^2/d_j$, the claimed isometry fails.","tokens_in":9831,"feed_emoji":"📍","tokens_out":15538,"duration_ms":125924,"temperature":0.7,"pith_summary":"Neumann eigenmaps (NeuMaps) are a landmark-based manifold-embedding method. The paper's central claim is that, when a subset of data points is treated as a subgraph, the Neumann eigenvectors of that subgraph give coordinates whose Euclidean distances reproduce the diffusion distance of a reflecting random walk on the subgraph. If the claim holds, a small landmark set can be embedded with the same geometric content as a full diffusion map while the discarded points contribute through reflections at the boundary. This matters because landmarking is the standard way to scale spectral embedding to large datasets, and NeuMap adds two properties: the removed points inform the embedding instead of being ignored, and out-of-sample points receive coordinates through a Nyström-type formula.","feed_headline":"Neumann eigenmaps keep diffusion geometry intact","feed_subtitle":"A reflecting random walk on a landmark subgraph gives coordinates that match its probability metric and extend to the full dataset.","key_machinery":"The central object is the renormalized Neumann Laplacian $L_S^N = L_S^D - B_S^T (T_S^\\delta)^{-1} B_S$, which combines the Dirichlet Laplacian on the landmark subgraph with a boundary term that enforces a vanishing discrete normal derivative at the removed vertices. The paper shows that $T_S^{-1/2}(I - L_S^N)T_S^{-1/2}$ is similar to a row-stochastic matrix $R$, the reflecting random walk transition matrix. The identity doing the work is the isometry formula in Theorem III.3, which equates the squared NeuMap distance with a degree-weighted sum of squared differences of reflecting-walk transition probabilities; the Neumann boundary condition also gives the closed-form Nyström-type extension used for out-of-sample points.","core_discovery":"The paper defines the Neumann Laplacian of a landmark subgraph $S$ as $L_S^N = L_S^D - B_S^T (T_S^\\delta)^{-1} B_S$, where $B_S$ records edges from $S$ to the removed vertices $\\delta S$ and $T_S^\\delta$ is their degree matrix. Its normalized eigenvectors define a row-stochastic transition matrix $R$ for the reflecting random walk on $S$, in which a walker moves from one landmark to another either directly or through a reflection off a removed vertex. The Neumann map $g_t^d(x_i)$ lists the scaled $t$-step coordinates of the left eigenvectors of $R$ at vertex $i$, starting from the second eigenvector. Theorem III.3 claims that, with $p_{ji}^t$ the reflecting-walk transition probabilities, the identity $\\sum_j (p_{ji}^t - p_{jk}^t)^2/d_j = \\|g_t(x_i)-g_t(x_k)\\|^2$ holds, so the embedding is isometric for the reflecting-walk diffusion distance. The same boundary condition provides an extension formula to removed points that is a multiple of the Nyström extension.","pith_inferences":["One testable extension is to include the discarded first coordinate explicitly: with the paper's left-eigenvector normalization that coordinate is degree-proportional, so adding it back may restore exact isometry on non-regular graphs.","The reflecting-walk interpretation suggests that NeuMap's stability should be largest when removed points are high-degree or metastable points, precisely the points whose removal distorts a plain diffusion map; varying the landmark-selection rule would test this.","The same renormalized Neumann construction could be applied with interior operators other than the Dirichlet Laplacian, defining reflecting-walk distances for alternative dynamics such as Schrödinger or transport operators."],"forward_implications":["NeuMap coordinates recover the diffusion distance of the reflecting random walk on the landmark subgraph, so the embedding is isometric in that probability metric.","The eigendecomposition is performed only on the landmark subgraph, giving a computational speedup that grows as the landmark set shrinks.","The Neumann boundary condition yields a Nyström-type extension, so removed data points receive coordinates without a second eigendecomposition.","In the digit example, NeuMap achieves higher cluster accuracy and normalized mutual information than the Roseland baseline on the same landmarks.","In the butane example, the first NeuMap eigenfunction tracks the dihedral angle with lower standard error than diffusion maps and stays stable when metastable-state or uniformly subsampled landmarks are removed."],"supporting_citations":[{"why":"Supplies the theory of Neumann spectra of subgraphs, including the vanishing-normal-derivative property (Lemma 8.1) that defines the boundary condition NeuMap uses.","marker":"[1]"},{"why":"Establishes diffusion maps and the diffusion-distance principle that NeuMap transposes to the reflecting random walk on landmarks.","marker":"[8]"},{"why":"Introduces the landmark-diffusion idea of letting the deleted points inform the embedding of the rest of the set, the construction NeuMap refines.","marker":"[21]"},{"why":"Identifies the Nyström extension that the Neumann boundary condition reproduces for out-of-sample points.","marker":"[45]"}],"fun_headline_variants":["Reflecting walk eigenmaps preserve diffusion distance","Landmark embedding matches reflecting random walk metric","Neumann eigenmaps extend Nyström to all data","Reflecting subgraph gives exact diffusion coordinates","NeuMaps: isometric landmark embedding for diffusion"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that throwing away the first Neumann eigenvector, as the method does, removes only a constant coordinate; with the paper's left-eigenvector normalization that coordinate is proportional to the vertex degree, which is constant only when all landmark vertices have equal degree.","fun_headline_variants_meta":{"raw":{"variants":["Reflecting walk eigenmaps preserve diffusion distance","Landmark embedding matches reflecting random walk metric","Neumann eigenmaps extend Nyström to all data","Reflecting subgraph gives exact diffusion coordinates","NeuMaps: isometric landmark embedding for diffusion"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0006,"raw_usage":{"total_tokens":2794,"prompt_tokens":928,"completion_tokens":1866,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":544,"completion_tokens_details":{"reasoning_tokens":1794}},"tokens_in":544,"tokens_out":1866,"duration_ms":13249,"temperature":1.0,"reasoning_tokens":1794,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T14:42:20.371339+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute both sides of the identity in Theorem III.3 for a small non-regular graph, for instance a path of three landmarks with one boundary vertex attached to the middle landmark. If the squared NeuMap distance does not equal $\\sum_j (p_{ji}^t-p_{jk}^t)^2/d_j$, the claimed isometry fails.","supporting_citations":[{"cited_title":"Spectral graph theory","cited_arxiv_id":null,"evidence_quote":"Supplies the theory of Neumann spectra of subgraphs, including the vanishing-normal-derivative property (Lemma 8.1) that defines the boundary condition NeuMap uses."},{"cited_title":"and Lafon, S., 2006","cited_arxiv_id":null,"evidence_quote":"Establishes diffusion maps and the diffusion-distance principle that NeuMap transposes to the reflecting random walk on landmarks."},{"cited_title":"and Wu, H.T., 2022","cited_arxiv_id":null,"evidence_quote":"Introduces the landmark-diffusion idea of letting the deleted points inform the embedding of the rest of the set, the construction NeuMap refines."},{"cited_title":"and Halevy, A","cited_arxiv_id":null,"evidence_quote":"Identifies the Nyström extension that the Neumann boundary condition reproduces for out-of-sample points."}],"review_version":1}