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Beyond Uniform Local Isometry and Topology: FactoMap for Disentangled Representations
Sohini Gupta, Bahareh Tolooshams
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Abstract
Abstract:Many disentanglement methods represent generative factors using Euclidean product coordinates, although the underlying factor spaces may wrap, collapse, or have position-dependent geometry. We introduce factor-space structure, combining factor domains, generator-induced identifications, and position-dependent scales to distinguish topologically equivalent spaces with different factor geometries. We show that statistically independent factors need not be geometrically separable: hue and scale produce effects that grow at different rates, yielding anisotropy that no fixed rescaling removes. We propose the Factor-Space Topographic Map (FactoMap), which learns interpretable prototypes indexed by a factor-space lattice. Topographic learning transfers the lattice's periodicity, collapses, and non-uniform extent to the representation. Experiments show that matching this structure preserves factor continuity and enables disentanglement of the underlying factors.
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- Source: https://arxiv.org/abs/2608.24762v1
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Symmetry and Geometry in Neural Representations Beyond Uniform Local Isometry and Topology: FactoMap for Disentangled Representations Sohini Gupta Affiliation: University of Alberta Affiliation: Alberta Machine Intelligence Institute (Amii) Bahareh Tolooshams Affiliation: University of Alberta Affiliation: Alberta Machine Intelligence Institute (Amii) Affiliation: Canada CIFAR AI Chair Abstract Many disentanglement methods represent generative factors using Euclidean product coordinates, although the underlying factor spaces may wrap, collapse, or have position-dependent geometry. We introduce factor-space structure, combining factor domains, generator-induced identifications, and position-dependent scales to distinguish topologically equivalent spaces with different factor geometries. We show that statistically independent factors need not be geometrically separable: hue and scale produce effects that grow at different rates, yielding anisotropy that no fixed rescaling removes. We propose the Factor-Space Topographic Map (FactoMap), which learns interpretable prototypes indexed by a factor-space lattice. Topographic learning transfers the lattice’s periodicity, collapses, and non-uniform extent to the representation. Experiments show that matching this structure preserves factor continuity and enables disentanglement of the underlying factors. 1 Introduction Disentangled representation learning seeks to recover the factors that generate a dataset, so that changing one factor of the world changes one coordinate of the representation while leaving the others fixed [1, 2, 3]. Such representations are appealing because their coordinates can support interpretation, generalization, fairness, and control [4, 5, 6, 7, 8, 9]. However, the factors underlying an arbitrary generative process cannot be recovered from unlabelled observations without additional assumptions [10]. Existing approaches impose inductive biases through statistics or geometry. Statistical methods encourage factorized representations [11, 12, 13, 14], while geometric ones constrain how distances vary between factors and observations [15, 16, 17]. A prominent geometric assumption is local isometry: equal-sized changes in different factor directions should produce equal-sized changes in the observation, up to a fixed global choice of units. This enables identifiability under suitable conditions [18, 19], and has motivated methods that favour flat or distance-preserving representations [20]. Local isometry, however, is not generally a property of the data-generating process. Different factors act through different mechanisms, and their effects can depend on the current factor configuration. For an object with independently varying hue and scale, changing hue affects its area, whereas changing scale primarily affects its boundary. Because area and perimeter grow at different rates, their relative effect vary as the object grows: the mismatch is position-dependent and cannot be removed by a fixed rescaling of the units. Statistical independence therefore does not imply geometric separability. Factors may be sampled independently while remaining geometrically coupled through the generator. Local geometry cannot determine whether a factor terminates or closes: an interval and a circle are locally indistinguishable, although a Euclidean coordinate must cut a periodic factor. Non-Euclidean topologies address this closure problem [21, 22, 23, 24], but topology [25] alone remains insufficient. A square, a planar disk, and a finite conical surface are homeomorphic [25], yet support different factor organizations. A circular factor may retain constant extent, shrink, or collapse at one point. These spaces therefore share a topology while differing in their factor identifications and local scale-structure; a representation encoding only dimension and topology cannot express these distinctions. Figure 1: FactoMaps: Factor-Space + Topology + Topography. A. Representative factor spaces and lattices; the disk and cone share topology but differ in factor-space geometry. B. Factor distributions and samples. C. FactoMap learns topographic prototypes on distinct factor-space structures. We introduce factor-space structure and the Factor-Space Topographic Map (FactoMap, Figure 1), a structured prototype learner designed to disentangle data by aligning lattice coordinates with the underlying factors, even when they do not admit uniform Euclidean geometry. The formalism combines topology with generator-induced identifications and position-dependent scales; FactoMap realizes this structure as a lattice of interpretable prototypes, learned through topographic cooperation [26, 27]. • Factor-space formalism. We separate factor domains, generator-induced identifications, and factor-wise scales, distinguishing homeomorphic spaces with different organizations or geometries. • Failure of uniform local isometry. We show that statistically independent hue and scale are geometrically coupled, producing position-dependent anisotropy that fixed rescaling cannot remove. • Factor-space prototype disentanglement. We propose FactoMap, whose supplied lattice distance encodes periodic closure, collapsed fibres, and non-uniform factor extent. Topographic learning organizes the prototypes to vary smoothly along factor-space paths, making the lattice coordinates a structured representation aligned with the underlying factors. • Controlled evaluation. In one- and two-factor settings, we compare matched and mismatched prototype domains to isolate how factor-space structure affects disentanglement. 2 Factor-Space Topographic Map (FactoMap) Factor-space structure. Topology specifies neither generator-induced identifications nor position-dependent factor scales. We therefore propose the following factor-space structure. Definition 2.1 (Factor-Space Structure). Let =∏i=1pZiZ= _i=1^pZ_i, with Zi∈I,S1Z_i∈\I,S^1\ and I=[0,1]I=[0,1], and let g:→ℝmg:Z ^m be piecewise continuously differentiable. Let i e_i denote the i-th coordinate direction, with increments along a circular factor taken modulo its period. The generator induces ≈g′⇔g()=g(′),ci():=∥∂g∂zi()∥2,g:=(c1,…,cp) z _g z g( z)=g( z ),\ c_i( z):=\| ∂ g∂ z_i( z)\|_2,\ c_g:=(c_1,…,c_p). We call g:=(,≈g,g) F_g:=(Z, _g, c_g) the factor-space structure induced by g, and ℱg:=/≈gF_g:=Z/\! _g, the realized factor space, equipped with the quotient topology. ci()c_i( z) measures the local data-space effect of factor i, i.e., for admissible δ→0δ→ 0, we have ‖g(+δi)−g()‖2=ci()|δ|+o(|δ|). \|g( z+δ e_i)-g( z) \|_2=c_i( z)|δ|+o(|δ|). (2.1) The pair (,≈g)(Z, _g) determines the topology of ℱgF_g, while g c_g records its factor-wise local scales on Z (see Section A.1). FactoMap models diagonal factor-wise scales; the HSV renderer additionally induces unmodelled cross-terms, analyzed in Appendix B. Beyond uniform Euclidean factor geometry. Some geometric approaches to disentanglement impose local isometry to a Euclidean factor domain [18, 15, 20, 16, 19]. Let g()=[∂g/∂z1,…,∂g/∂zp] J_g( z)=[∂ g/∂ z_1,…,∂ g/∂ z_p] and ()=g()⊤g() G( z)= J_g( z) J_g( z). Up to a fixed global scale, local isometry to uniform Euclidean factor coordinates requires ()=C2 G( z)=C^2 I. This fails when factor directions have nonzero cross-terms or when their diagonal scales are unequal or position-dependent. FactoMap addresses the latter by allowing its lattice geometry to vary with the scale functions in Definition 2.1. Consider hue h∈S1h∈ S^1 and scale s∈[smin,smax]s∈[s_ ,s_ ], smin>0s_ >0, on a black background: g(h,s)xy=mxy(s)(h)g(h,s)_xy=m_xy(s)\, ρ(h). The exponent γ depends on the rendering model; a fixed-width transition gives γ=1/2γ= 12. Proposition 2.1 (Scale-dependent anisotropy). Suppose that msm_s and ρ are continuously differentiable and, for all s∈[smin,smax]s∈[s_ ,s_ ] and h∈S1h∈ S^1, ‖ms‖2=as,‖∂sms‖2=bsγ,‖(h)‖2=q,‖′(h)‖2=v\|m_s\|_2=a\,s,\ \| _sm_s\|_2=b\,s^γ,\ \| ρ(h)\|_2=q,\ \| ρ (h)\|_2=v, where a,b,q,v>0a,b,q,v>0. The constant-radius condition implies ⟨(h),′(h)⟩=12∂h‖(h)‖22=0 ρ(h), ρ (h) = 12 _h\| ρ(h)\|_2^2=0. Hence, the hue and scale directions are orthogonal and cs(h,s)=bqsγ,ch(h,s)=avs,κ(s):=ch(h,s)cs(h,s)=avbqs1−γ.c_s(h,s)=bq\,s^γ, c_h(h,s)=av\,s, κ(s):= c_h(h,s)c_s(h,s)= avbq\,s^1-γ. (2.2) Hence, (h,s)=diag(a2v2s2,b2q2s2γ) G(h,s)=diag(a^2v^2\,s^2,b^2q^2\,s^2γ), and κ(smax)/κ(smin)=(smax/smin)1−γ κ(s_ )κ(s_ )=( s_ s_ )^1-γ. No fixed diagonal re-weighting makes =C2 G=C^2 I throughout a non-degenerate scale interval. Hence, statistically independent factors need not be geometrically separable: the observation-space extent of hue varies with scale. A uniform cylinder cannot represent this dependence, whereas a matched lattice varies the extent of its cyclic axis along the scale axis (see Section A.2). Factor-space topographic map. We propose FactoMap, a structured prototype learner to realize a factor-space structure as an interpretable topographic representation. It has three components: a factor-space lattice, a data-space prototype field, and a topographic learning objective. Factor-space lattice. For a supplied structure F, let (Λ,d)( _ F,d_ F) be a finite lattice with an analytic distance chosen to realize that structure (see Section A.3). Interval and circular factors determine whether axes terminate or wrap; identifications determine which fibres meet or collapse; and the scale functions inform their relative local extent. This finite realization need not be unique. Prototype representation. Each site k∈Λk∈ _ F indexes a learnable prototype k∈ℝm w_k ^m, defining W:Λ→ℝmW: _ F ^m, k↦k w_k. FactoMap encodes an observation by its best-matching lattice coordinate, qW():=argmink∈Λ‖−k‖22.q_W( x):= _k∈ _ F\| x- w_k\|_2^2. (2.3) Thus, qW()q_W( x) is a factor-space-indexed discrete coordinate. Topographic learning. For neighbourhood width σt _t, define Hσt(j,k)=exp[−d(j,k)2/(2σt2)]H_ _t(j,k)= [-d_ F(j,k)^2/(2 _t^2)]. We learn the prototypes with the self-organizing-map objective [26] ℒt(W)=[∑k∈ΛHσt(qW(),k)‖−k‖22].L_t(W)=E_ x\! [ _k∈ _ FH_ _t\! (q_W( x),k )\| x- w_k\|_2^2 ]. (2.4) The neighbourhood term encourages nearby factor-space sites to represent nearby observations (see Section A.4). (Λ,d,W,qW)( _ F,d_ F,W,q_W) defines the learned FactoMap: the lattice encodes the supplied topology and geometry, while topographic learning realizes them in observation-space prototypes. 3 Results We introduce FactoShapes, a synthetic dataset with four controlled generative factors: object hue, scale, and horizontal and vertical position, and evaluate FactoMap on it (see Appendix B). Topology matching removes the cut in a periodic factor. For the interval-valued scale factor, a line-lattice FactoMap learns an ordered sequence of prototypes whose object size changes smoothly along the lattice (Figure 2A). Finite differences on the rendered images show that the local scale speed increases approximately as cs(s)∝sc_s(s) s, consistent with the analytical fixed-width rendering model. Applying the same open lattice to hue introduces a cut: hue values near 00 and 11 are adjacent in factor space, but their best-matching sites lie at opposite lattice ends. A ring lattice reduces this separation and produces a continuous cycle of hue prototypes. Thus, local continuity does not determine global closure; periodicity must be encoded in the factor-space lattice. The prototype-distance profiles show how the structures are realized. For scale, the nonlinear cumulative distance is consistent with a position-dependent cs(s)c_s(s). Along the hue ring, distance from a reference prototype increases toward the antipode and then decreases, as expected for a cyclic factor. Figure 2: FactoMap disentangles factors when topology and local scale match their factor-space structure. A. A line lattice represents scale continuously. For periodic hue, an open lattice separates adjacent endpoint hues (Δlattice=39 _lattice=39), whereas a ring restores their adjacency (Δlattice=1 _lattice=1). B. With hue and scale varied jointly, the measured image-space speeds agree with the analytical laws ch∝sc_h s and cs∝sc_s s. A rectangular lattice mixes the two factors, whereas the matched cone-structured lattice wraps hue and varies its extent with scale, yielding factor-aligned prototypes. Independent factors exhibit position-dependent anisotropy. When hue and scale vary jointly, their measured image-space speeds follow different power laws (Figure 2B, top). Power-law fits give chmeas(s)∝s1.05c_h^meas(s) s^1.05 and csmeas(s)∝s0.56c_s^meas(s) s^0.56, close to the analytical predictions chth(s)∝sc_h^th(s) s and csth(s)∝s0.5c_s^th(s) s^0.5 from Proposition 2.1. Consequently, κ(s)=ch(s)/cs(s)∝s1.05−0.56=s0.49κ(s)= c_h(s)c_s(s) s^1.05-0.56=s^0.49. Over s∈[0.75,1.5]s∈[0.75,1.5], the fitted anisotropy therefore changes by κ(smax)/κ(smin)=20.49≈1.404 κ(s_ )κ(s_ )=2^0.49≈ 1.404, in close agreement with the analytical prediction 2≈1.41 2≈ 1.41. Thus, no fixed rescaling matches both directions across scales: independently sampled hue and scale remain geometrically coupled by the generator. The matched factor-space inductive bias enables hue-scale disentanglement. As predicted by Proposition 2.1, the two-dimensional lattice must represent unequal, position-dependent factor scales: its cyclic circumference must vary with ch(s)∝sc_h(s) s, while displacement along its scale direction must reflect cs(s)∝sc_s(s) s. The measured power laws above closely match these analytical rates. A rectangular lattice neither wraps hue nor varies its cyclic extent; empirically, its prototypes mix hue and scale rather than aligning the two factors with separate coordinates (Figure 2B, bottom). Table 1: Disentanglement with matched cone and mismatched grid. Lattice InfoM ↑ InfoE ↑ InfoC ↑ ✔ Cone 0.953 0.822 0.950 ✘ Grid 0.272 0.544 0.025 FactoMap uses a supplied cone lattice over the sampled interval: hue wraps around the cyclic coordinate, and the circumference of its fibres varies with scale (note: because smin>0s_ >0, the experimental lattice contains no collapsed fibre; hence, practically the prototypes are on a conical frustum. Collapse at s=0s=0 remains a formal extension illustrated in Figure 1). The analytic distance incorporates the factor-wise scales, using ch(s)c_h(s) for the cyclic extent and cs(s)c_s(s) for displacement along scale. The learned prototypes consequently organize hue around the cyclic coordinate and scale along the non-periodic coordinate. In this experiment, the mismatched rectangular lattice fails to disentangle hue and scale, whereas supplying the matched cone structure yields a factor-aligned prototype representation. This qualitative organization is confirmed quantitatively in Table 1 using InfoMEC [9]; the matched cone lattice substantially improves disentanglement over the mismatched grid. Acknowledgments and Disclosure of Funding S.G. and B.T. would like to thank Valérie Costa for helpful discussions. S.G. and B.T. acknowledge support of Natural Sciences and Engineering Research Council of Canada (NSERC), RGPIN-2026-05959, and funding from the Canada CIFAR AI Chairs Program. References [1] Y. Bengio, A. Courville, and P. Vincent (2013) Representation learning: a review and new perspectives. IEEE transactions on pattern analysis and machine intelligence 35 (8), p. 1798–1828. Cited by: §1. [2] I. Higgins, D. Amos, D. Pfau, S. Racaniere, L. Matthey, D. Rezende, and A. Lerchner (2018) Towards a definition of disentangled representations. External Links: 1812.02230, Link Cited by: §1. [3] G. Desjardins, A. Courville, and Y. 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We denote scalars as non-bold-lower-case a, vectors as bold-lower-case a, and matrices as upper-case letters A. ∥⋅∥2\|·\|_2 denotes the ℓ2 _2 (Euclidean) norm. The generator g:→ℝmg:Z ^m maps factor configurations ∈ z to observations x. We write g=(,≈g,g) F_g=(Z, _g, c_g) for the factor-space structure induced by g and ℱg=/≈gF_g=Z/\! _g for its realized factor space. For a supplied structure F, Λ _ F denotes its finite lattice realization and d_ F its analytic lattice distance. Each site k∈Λk∈ _ F indexes a prototype k∈ℝm w_k ^m, and W:Λ→ℝmW: _ F ^m denotes the prototype field. The encoder qW()q_W( x) returns the best-matching lattice site, Hσt(j,k)H_ _t(j,k) is the topographic neighbourhood kernel at iteration t, and ℒt(W)L_t(W) and ℒ^t(W) L_t(W) denote the population and minibatch objectives, respectively. Finally, p is number of factors, and K:=|Λ|K:=| _ F| is the number of prototypes. A.1 Remarks on the factor-space structure The realized factor space ℱg=/≈gF_g=Z/\! _g carries the quotient topology induced by the canonical projection π:→ℱgπ:Z _g. The scale functions are defined on Z and need not descend to the quotient: descent would require ci()=ci(′)whenever≈g′.c_i( z)=c_i( z ) z _g z . (A.1) This distinction matters at collapsed fibres, where the generator can identify an entire coordinate fibre while the remaining scale functions still vary along that fibre. The complete pullback metric [28] is Gij()=⟨∂g∂zi(),∂g∂zj()⟩,G_ij( z)= ∂ g∂ z_i( z), ∂ g∂ z_j( z) , (A.2) whose diagonal entries satisfy Gii=ci2G_i=c_i^2. The present FactoMap construction uses these diagonal scales and analytic separable lattice distances; it does not represent nonzero off-diagonal terms. Neither g c_g nor the quotient topology uniquely determines a global lattice distance. The choice of (Λ,d)( _ F,d_ F) is therefore a supplied finite realization of the factor-space structure and is part of the model design. A.2 Derivation of the hue-scale metric See 2.1 Proof. Because the background is black, the generator separates into a spatial mask and a colour vector: ∂g∂h=ms′(h),∂g∂s=(∂sms)(h). ∂ g∂ h=m_s\, ρ (h), ∂ g∂ s=( _sm_s)\, ρ(h). (A.3) Taking norms gives ch=‖ms‖2‖′(h)‖2=avs,cs=‖∂sms‖2‖(h)‖2=bqsγ.c_h=\|m_s\|_2\| ρ (h)\|_2=av\,s, c_s=\| _sm_s\|_2\| ρ(h)\|_2=bq\,s^γ. (A.4) Moreover, ⟨∂g∂h,∂g∂s⟩=⟨ms,∂sms⟩⟨′(h),(h)⟩=0, ∂ g∂ h, ∂ g∂ s = m_s, _sm_s ρ (h), ρ(h) =0, (A.5) because constant colour magnitude implies ⟨′(h),(h)⟩=0 ρ (h), ρ(h) =0. Substitution into =g⊤g G= J_g J_g proves the stated metric and scale ratio. A fixed diagonal coordinate re-weighting multiplies chc_h and csc_s by constants and therefore cannot remove their positional dependence. In particular, the hue-hue entry remains proportional to s2s^2, so the metric cannot equal C2C^2 I on a non-degenerate scale interval. ∎ Factor-wise reparameterization. For coordinate changes h=η(u)h=η(u) and s=σ(t)s=σ(t), the transformed scales are c~u=avσ(t)|η′(u)|,c~t=bqσ(t)γ|σ′(t)|. c_u=av\,σ(t)|η (u)|, c_t=bq\,σ(t)^γ|σ (t)|. (A.6) The first cannot be constant on a non-degenerate scale interval because its dependence on t cannot be cancelled by η′(u)η (u). Hence no factor-wise reparameterization makes the metric uniformly Euclidean. A scale reparameterization can equalize the two diagonal entries point-wise, but their common value remains position-dependent, yielding a conformal rather than a uniformly isometric metric. Dependence on the mask model. The exponent γ is renderer-dependent. For a smooth self-similar mask ms()=ϕ(/s)m_s( r)=φ( r/s), a change of variables gives ‖ms‖2∝s,‖∂sms‖2∝1,\|m_s\|_2 s, \| _sm_s\|_2 1, (A.7) so γ=0γ=0. For a smooth transition of fixed spatial width, the changing pixels occupy a band whose area grows with the object perimeter, giving ‖∂sms‖22∝s,‖∂sms‖2∝s,\| _sm_s\|_2^2 s, \| _sm_s\|_2 s, (A.8) and hence γ=12γ= 12. A hard binary mask is not differentiable in L2L^2 with respect to s; finite-step lattice distances nevertheless remain well defined. The measured exponent γ≈0.56γ≈ 0.56 is close to the fixed-width prediction but should be interpreted as a property of the rendering pipeline. The theoretical conclusion does not depend on the precise exponent: in every smooth model above, the hue scale depends on the scale factor, so the induced geometry is not a uniform Euclidean product in the original factor coordinates. Collapse at zero scale. If the model is extended by m0≡0m_0≡ 0, then g(h,0)=0g(h,0)=0 for every h∈S1h∈ S^1. Thus, ≈g _g identifies the entire hue fibre at s=0s=0, while ch(h,s)=avs→0c_h(h,s)=av\,s→ 0. The identification and vanishing scale record the same collapse globally and locally. A.3 Analytic lattice distances FactoMap receives the finite lattice and its distance as inputs: (Λ,d)( _ F,d_ F). No topology selection, identification estimation, or scale estimation is performed in the present work. For a regular product lattice ∏i=1p0,…,Si−1 _i=1^p\0,…,S_i-1\, define ρi(a,b)=|a−b|,Zi=I,min|a−b|,Si−|a−b|,Zi=S1. _i(a,b)= cases|a-b|,&Z_i=I,\\[2.0pt] \|a-b|,S_i-|a-b|\,&Z_i=S^1. cases (A.9) A weighted product distance is d(,)2=∑i=1pλi2ρi(ji,ki)2,d_ F( j, k)^2= _i=1^p _i^2 _i(j_i,k_i)^2, (A.10) where λi>0 _i>0 fixes the lattice unit of factor i. Open axes give lines and grids; wrapped axes give rings, cylinders, and tori. For collapsed or non-uniform structures, (A.10) is replaced by the analytic distance used by the corresponding lattice. These distances must specify both the collapse implementation and the dependence of one factor’s extent on another. Finally, because the neighbourhood depends on d/σtd_ F/ _t, a global rescaling of the lattice distance is equivalent to rescaling the neighbourhood schedule. Distance normalization and the σt _t schedule must therefore be specified together. A.4 Optimization For each mini-batch ℬtB_t, we compute bn=argminj∈Λ‖n−j‖22,n∈ℬt,b_n= _j∈ _ F\| x_n- w_j\|_2^2, x_n _t, (A.11) and hold bnb_n fixed during the subsequent gradient step. Conditional on these assignments, the minibatch objective is ℒ^t(W)=1|ℬt|∑n∈ℬt∑k∈ΛHσt(bn,k)‖n−k‖22, L_t(W)= 1|B_t| _ x_n _t _k∈ _ FH_ _t(b_n,k)\| x_n- w_k\|_2^2, (A.12) with prototype gradient ∇kℒ^t=2|ℬt|∑n∈ℬtHσt(bn,k)(k−n). _ w_k L_t= 2|B_t| _ x_n _tH_ _t(b_n,k)( w_k- x_n). (A.13) We update the prototypes with Adam [29] and recompute the BMUs after every update; no gradient is propagated through the argmin . The experimental details specify the Adam parameters, batch size, number of updates, prototype initialization, and the complete schedule for σt _t annealing. Appendix B Experimental Details B.1 Datasets FactoShapes. We introduce FactoShapes, a synthetic image dataset of resolution 64×64×364× 64× 3 RGB images (Figure 3). The generative factors are v=(object_hue,scale,posx,posy)v=(object\_hue,\,scale,\,pos_x,\,pos_y), so that the factor space is S1×I3S^1× I^3: hue is periodic (HSV hue in [0,1)[0,1) at fixed saturation ⟨S⟩ S and value ⟨V⟩ V ), while scale and the two position factors are intervals, s∈[smin,smax]s∈[s_ ,s_ ] and px,py∈[−p⋆,p⋆]p_x,p_y∈[-p ,p ]. The bound p⋆p is obtained by a numerical solver as the largest offset for which the projected object remains fully inside the frame at the largest scale smaxs_ , guaranteeing that no sample is clipped and that the four factors are independently variable over the whole product range. Sampling. FactoShapes supports three modes. In grid mode the dataset is the Cartesian product of nh×ns×nx×nyn_h× n_s× n_x× n_y factor values, giving N=⟨N⟩N= N images with integer factor classes, matching the Shapes3D [30] convention. In random-binned mode factor values are drawn uniformly from the same discrete grid, so that each image is a random lattice point and factor classes remain defined. In continuous mode each factor is drawn i.i.d. with a choice of uniform or Gaussian distribution from its continuous range; no image is repeated, and the integer class array is set to −1-1 throughout. Datasets are written in 3dshapes format: images of shape (N,64,64,3)(N,64,64,3) uint8, labels of shape (N,4)(N,4) float64, latents_classes of shape (N,4)(N,4) int64, together with attributes recording the factor names and ranges. Figure 1B shows the distribution and samples of the dataset when hue is in [0,1)[0,1) with S=V=1, s∈[0.75,1.5] and px,py∈[0.556,0.556]s∈[0.75,1.5] and p_x,p_y∈[0.556,0.556]. Hue and scale as the two varying factors. We vary object_hue (HSV hue in [0,1)[0,1) with S=V=1S=V=1) and scale (s∈[0.75,1.5]s∈[0.75,1.5]) while keeping the posxpos_x and posypos_y fixed. We generate 100,000100,000 samples in the continuous mode with each varying factor drawn from i.i.d. uniformly for our experiments. Figure 3 shows the sampling distribution with visuals of generated sample. Figure 3: FactoShapes Dataset. Sampling distribution (uniform) and visuals of samples for object_hue and scale as varying factors. HSV-induced metric cross-term. Proposition 2.1 assumes a colour path with constant RGB norm about the black background, whereas the experiments use HSV(h)=HSVtoRGB(h,1,1) ρ_HSV(h)=HSVtoRGB(h,1,1) with normalized hue h∈[0,1)h∈[0,1). This path has non-constant RGB norm and is differentiable only within each of its six linear sectors. In the first sector, h∈[0,16]h∈[0, 16], it is (h)=(1,6h,0)=(1,τ,0),τ:=6h∈[0,1]. ρ(h)=(1,6h,0)=(1,τ,0), τ:=6h∈[0,1]. (B.1) The remaining sectors have the same form up to a channel permutation and orientation. Away from the sector boundaries, ‖(h)‖2=1+τ2,‖′(h)‖2=6,|⟨(h),′(h)⟩|=6τ.\| ρ(h)\|_2= 1+τ^2, \| ρ (h)\|_2=6, | ρ(h), ρ (h) |=6τ. (B.2) Consequently, the normalized colour contribution to the cross-term is |⟨,′⟩|‖2‖′‖2=τ1+τ2. | ρ, ρ |\| ρ\|_2\| ρ \|_2= τ 1+τ^2. (B.3) For g(h,s)=ms(h)g(h,s)=m_s ρ(h), the metric cross-term factorizes as Ghs(h,s)=⟨ms,∂sms⟩⟨′(h),(h)⟩,G_hs(h,s)= m_s, _sm_s ρ (h), ρ(h) , (B.4) and is therefore generally nonzero. Under the leading-order mask model, however, the diagonal scales retain the scale dependence ch(h,s)=6as∝s,cs(h,s)=b1+τ2sγ∝sγc_h(h,s)=6as s, c_s(h,s)=b 1+τ^2\,s^γ s^γ (B.5) almost everywhere. Thus, the supplied lattice matches the factor topology and the power-law dependence on scale, but not the HSV-induced cross-term or the hue-dependent prefactor of csc_s. The results consequently demonstrate the benefit of this partial geometric match rather than general handling of non-diagonal metrics. B.2 Training and model parameters Training. Training uses a single step counter, while step < num_steps. We take num_steps=60,000 num\_steps=60,000 updates at batch size B=128B=128, optimized with Adam at initial learning rate η0=0.1 _0=0.1 . We use the inbuilt ExponentialLR learning rate scheduler function with γ=e-1/num_stepsγ=e -1/num\_steps . The initial neighborhood radius is not set explicitly; the model initializes it to half its own topological diameter. The radius σ0 _0 exponentially decays to σT=1.0 _T=1.0 in the first 65% of the training steps and is held constant thereafter. Prototype field. For the cone FactoMap we specify the lattice size to be 40×2040× 20 with prototypes W∈ℝK×12,288W∈R^K× 12,288 where K=800K=800 here. Knowing the fact that the rows are wrapped we have an increased number on the rows. The lattice carries cone metric: a ring at radius r has circumference 2πkr2π kr with k=0.243k=0.243, and column j sits at radius rj=r0+jr_j=r_0+j with r0=18.4r_0=18.4, so radii run from 18.418.4 at the inner rim to 37.437.4 at the outer rim. The cone is used rather than a plain grid because an open rectangular grid cannot represent periodic hue without introducing a seam; wrapping the hue axis removes this artificial boundary. A cylinder is neither suitable as the generator’s induced metric is not a product metric: the hue and scale coefficients scale as ch∝s1.05c_h s^1.05 and cs∝s0.56c_s s^0.56, so the arc length of a full hue circle relative to radial spacing grows with scale. Forming a cone helps to preserve the relation. Appendix C Additional Visualizations We learn grid and cone prototype lattices on FactoShapes (Figure 3), where hue and scale are the only varying factors. On the grid, hue and scale don’t run systematically along the lattice axes, so neither factor is recoverable from any axis direction. The failure is informative: the learned prototypes still wrap in hue, indicating that the map is trying to close a cycle the grid cannot support. Matching this cyclic structure requires a lattice with a circular direction, and the taper of a cone additionally accommodates the growth of the hue metric with scale. On the cone lattice, hue aligns with the circular direction and scale with the axial direction (Figure 4), and Table 1 confirms this with InfoMEC. Figure 4: Visualizations of learned prototypes in a grid lattice and cone lattice. The cone is cut to unfold the lattice to visualization the alignment of prototypes.