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A Regularity Condition Under Which a CPAB Transformation is a Diffeomorphism
Amit Aflalo, Oren Freifeld
TL;DR
CPAB transformations are useful compact flows, but they are not necessarily diffeomorphisms despite being bi-Lipschitz homeomorphisms. This paper gives sufficient regularity conditions for diffeomorphism and examines their behavior in 1D, 2D, and 3D. The 3D argument guarantees diffeomorphism under its condition, though its probability-one applicability to standard tessellations is not established.
Problem
CPAB transformations are bi-Lipschitz homeomorphisms but are not necessarily diffeomorphisms, a gap relevant to spatial modeling requiring differentiable, invertible, topology-preserving deformations.
Source: §1 · pp. 1–2Method
The paper identifies sufficient regularity conditions for CPAB flows, extending the 2D differentiability argument to 3D when interior trajectories spend zero time on interfaces where the velocity derivative is discontinuous.
Source: §4 · p. 13Results
In 1D, zero boundary conditions and matching-slope interface zeros are compatible with diffeomorphism; 2D edge zeros have probability zero under free endpoints, while the 3D probability-one claim remains unproved.
Source: §2 · pp. 5–6, §3 · p. 13, §4 · p. 13Takeaways & Limitations
Under the stated trajectory/interface condition, the 3D flow map is a diffeomorphism from U_T onto its image for every T > 0.
Source: §4 · p. 13For standard five-tetrahedron subdivisions, probability-one regularity under joint-density sampling is expected but not established.
Source: §4 · p. 13
Abstract
from arXiv · showhide
Continuous Piecewise-Affine Based (CPAB) transformations, introduced by Freifeld et al. (2015, 2017), are flows obtained by integrating continuous piecewise-affine (CPA) velocity fields. These expressive transformations have proven useful in applications including nonlinear temporal warping and trainable activation functions in 1D, as well as nonrigid spatial deformations in 2D and 3D. While computational costs were not an issue in 1D, they originally limited the practical expressiveness of CPAB transformations in higher dimensions, especially in 3D. Our recent work (Aflalo et al., 2026) removed this bottleneck, enabling efficient optimization with much finer tessellations and substantially greater expressiveness. There is, however, a mathematical subtlety that was overlooked in the original CPAB papers (Freifeld et al., 2015, 2017). CPAB transformations are always bi-Lipschitz homeomorphisms, implying that both the transformations and their inverses are differentiable almost everywhere. They are not, however, necessarily diffeomorphisms. This note remedies the omission by identifying sufficient regularity conditions under which a CPAB transformation is guaranteed to be a diffeomorphism. We explain how differentiability can fail and how the proposed conditions prevent this failure. In 1D and 2D, we show that these conditions can be easily enforced and, for commonly used tessellations, hold with probability one under standard sampling assumptions. We expect analogous behavior in 3D.
1 Introduction
CPAB offers a compact, expressive alternative to dense-grid transformations, but its flows are not always diffeomorphisms. This note identifies sufficient regularity conditions and examines their applicability across dimensions.
1 Introduction
- Dense-grid diffeomorphism methods scale their parameter count with domain resolution, causing high memory use, slow optimization, and difficult optimization landscapes in modern 2D and especially 3D settings.The note also describes a concurrent gradient-computation improvement that enables efficient optimization of fine-tessellation CPAB diffeomorphisms, including in 3D [1].Source: §1 · pp. 1–2
- CPAB integrates continuous piecewise-affine velocity fields over tessellated cells, yielding compact parameterization while capturing rich nonlinear deformations.The approach has been used in 1D alignment and activation functions, parameter-efficient fine-tuning, and 2D vision and generative modeling [19] [5] [17] [23] [18] [14] [22] [26] [13].Source: §1 · pp. 1–2
- CPAB flows are bi-Lipschitz homeomorphisms but are not necessarily diffeomorphisms; the note provides a sufficient regularity condition and examines its practical prevalence.The authors say the condition usually holds in 1D and 2D and expect similar behavior in 3D.Source: §1 · p. 2
- The analysis focuses on sufficient conditions guaranteeing differentiability of CPAB flows and whether they hold generically for commonly used tessellations.It distinguishes this setting—continuous, Lipschitz CPA fields—from related piecewise-smooth dynamical-systems literature that also studies discontinuous vector fields.Source: §1 · p. 2
2 The 1D Case
In 1D, a CPAB flow is a diffeomorphism when velocity zeros do not coincide with interior cell boundaries where adjacent slopes differ. The analysis distinguishes this failure case from several harmless zeros and shows the condition holds almost surely under specified sampling or perturbation procedures.
2 The 1D Case
- Where velocity is nowhere zero, an integral-coordinate representation yields a continuous positive flow derivative even across interfaces, and the inverse is differentiable.The derivative of the velocity itself may be discontinuous at crossed cell boundaries.Source: §2 · p. 4
- The nonvanishing argument applies locally at any starting point with nonzero velocity, even if zeros occur elsewhere, because trajectories cannot reach a zero in finite time.The trajectory stays in the maximal interval around its starting point on which velocity is nonzero.Source: §2 · p. 4
- A zero at a slope-mismatched interface makes one-sided flow derivatives differ, so the map remains bi-Lipschitz but is not differentiable there.For v(p) = |p|, the one-sided derivatives are e^-1 and e.Source: §2 · p. 7, Figure 6 · p. 7
- Zeros inside cells, at matched-slope interfaces, and at domain endpoints are compatible with diffeomorphic flows; nonzero velocity also permits slope kinks.At a matched-slope interface, the adjacent pieces define the same local affine function.Source: §2 · pp. 5–7
- In 1D, nonzero velocity at every interior cell boundary with differing adjacent slopes suffices for a C1 diffeomorphism.The transformation and its inverse are continuously differentiable; endpoint derivatives use the usual one-sided interpretation.Source: §2 · pp. 6–7
- Absolutely continuous vertex-velocity sampling or arbitrarily small Gaussian perturbations guarantee the 1D condition almost surely; gradient optimization is expected, not guaranteed, to satisfy it.The perturbations can preserve prescribed boundary conditions.Source: §2 · p. 8
3 The 2D Case
In 2D, the paper gives sufficient edge conditions for CPAB flows to be diffeomorphisms and shows these conditions hold almost surely for commonly used tessellations under standard sampling assumptions. It also explains how tangential motion along an interface can make a flow nondifferentiable despite a nonzero velocity field.
3 The 2D Case
- The nonzero field v(x, y) = (1, |y|) still produces a nondifferentiable flow because it is tangent to y = 0 and trajectories remain on that interface.The vertical component has unequal one-sided derivatives there, e^-1 and e.Source: §3 · pp. 8–9, Figure 8 · p. 9
- For interior trajectories, the two edge conditions—nonzero velocity and a normal component not identically zero—ensure a C1 diffeomorphism.The result applies to initial conditions whose trajectories remain in the domain interior during the time interval considered.Source: §3 · pp. 9, 10, 12
- The sufficient conditions are stronger than necessary: matching affine velocity fields across an interface cause no regularity issue even if those conditions fail.The paper adopts the stronger conditions because they are simple to verify and hold almost surely under standard sampling.Source: §3 · p. 10
- Under the edge conditions, every interior trajectory spends zero time on cell interfaces, enabling differentiation of the flow with respect to its initial condition and of its inverse.The resulting flow is a diffeomorphism onto its image.Source: §3 · pp. 10–12
- On the specified four-triangle-per-square grids, the edge conditions hold almost surely under absolutely continuous sampling when every interior edge has a free-velocity endpoint.The probability-one claim applies simultaneously across the finite set of interior edges.Source: §3 · pp. 12–13
- With zero boundary velocities, the interior maps onto itself and is diffeomorphic, while the closed-domain extension is the identity on the boundary; derivative continuity need not extend to boundary vertices.The transformation remains a bi-Lipschitz homeomorphism on the closed domain.Source: §3 · p. 13
4 The 3D Case
In 3D, the 2D differentiability argument extends under a condition that trajectories spend zero time on interior interfaces where the velocity derivative is discontinuous. For standard five-tetrahedron grids, the authors expect this condition to hold with probability one under joint-density sampling, but do not establish that result.
4 The 3D Case
- Under the zero-interface-time condition, every flow map Φ(T) for T > 0 is a diffeomorphism onto its image in the 3D domain.The domain is tessellated into tetrahedra with triangular interfaces; the proof extends the 2D differentiability argument.Source: §4 · p. 13
- In 3D, interface-tangent points may form an entire line rather than an isolated point, making genericity analysis more involved than in 2D.Source: §4 · p. 13
- For standard five-tetrahedron rectangular-grid subdivisions, including zero-normal-velocity boundary constraints, probability-one regularity is expected under joint-density sampling but is not proved.A Gaussian distribution is given as an example of a distribution with a joint density.Source: §4 · p. 13
5 The mistake in [10, 11]
The paper corrects the unqualified claim in the original CPAB papers that every transformation is a diffeomorphism: CPAB transformations are always bi-Lipschitz homeomorphisms, but diffeomorphism requires sufficient regularity. In 1D and the 2D tessellations considered, the conditions are easy to enforce and hold with probability one under joint-density sampling; analogous behavior is expected in 3D.
5 The mistake in [10, 11]
- The original CPAB papers incorrectly claimed every transformation is a diffeomorphism, relying on the false implication that Lipschitz velocity guarantees a differentiable flow.Source: §5 · pp. 13–14
- The original invertibility argument remains valid: CPAB transformations are always bi-Lipschitz homeomorphisms with almost-everywhere differentiable inverses, and sufficient regularity makes them diffeomorphisms.Source: §5 · p. 14
- In 1D and the 2D tessellations considered, the conditions are easy to enforce and hold with probability one under joint-density sampling; analogous behavior is expected in 3D.Source: §5 · p. 14