Conditioning Degenerate Diffusion Models

1 Introduction

• Conditioning is framed using data from an unconditioned process together with a terminal condition; at time t, the available data are the state history and condition.The setup includes an SDE with potentially rank-deficient diffusion and terminal condition Y = G(XT).Source: §1 · p. 1
• When the conditional likelihood is smooth, the Doob h-transform gives a Brownian shift using the score of that likelihood.The induced state drift is σσ⊤∇x log h, and the shift has a minimum-entropy characterization.Source: §1 · p. 1
• Degeneracy can make the state filtration differ from the driving Brownian filtration, complicating construction of a state-adapted closed-loop control.Such a control avoids requiring information beyond the state path and terminal condition.Source: §1 · pp. 1–2
• For identity terminal conditions, the terminal conditional law is a singular Dirac measure rather than a smooth density.This limits the smooth-likelihood setting used by score-based guidance.Source: §1 · p. 2

2.1 Related Work

• The paper uses the predictable representation property to derive a closed-loop Föllmer drift that minimizes energy among admissible shifts.It also relaxes a strict-positivity condition in prior causal optimal transport treatment through localization.Source: §2.1 · p. 2
• Existing approaches to degenerate or nonsmooth conditioned diffusions still impose density smoothness or coefficient regularity conditions.Related work includes bridge and manifold-based models, as well as Malliavin-calculus and other approaches.Source: §2.1 · p. 2

2.2 Contributions

• The predictable-representation result for degenerate diffusions is extended to random initializations.The earlier result is attributed to Üstünel (2019).Source: §2.2 · p. 2
• Local regression losses require neither a density nor a score function and identify the unique minimum-energy Girsanov shift under stated conditions.The conditions are local square integrability of the conditional Radon–Nikodym derivative and a state-feedback factorization.Source: §2.2 · p. 2
• The paper formulates the loss as a control problem and reports experiments varying ε and an image-generation application.The stated demonstrations are in Subsections 5.2 and 5.3.Source: §2.2 · pp. 2–3

3 Notation

• F_t(X) denotes the usual P-augmentation of the state-history sigma-field through time t, while F^Q_t(X) is its Q-augmentation.The raw state filtration is generated by X_s for 0 ≤ s ≤ t.Source: §3 · p. 3
• The notation L²_a specifies predictable processes with finite expected integrated squared norm, with variants restricting the time interval or requiring the condition locally before T.The local space requires finiteness on every interval [0,r] with r < T.Source: §3 · p. 3
• The projection Π(a(t)) is onto the range of a(t)⊤, and P_t(X) projects onto the range of σ(t,X_t)⊤.The notation also uses the Moore–Penrose pseudoinverse and Frobenius norm.Source: §3 · p. 3

4 Loss Functions for Degenerate Conditioned Diffusions

• The minimum-entropy argument uses pathwise uniqueness, while boundedness can be replaced by localized moment assumptions.Global Lipschitz continuity with linear growth is given as a standard sufficient condition for a unique strong solution.Source: §4 · p. 3
• The loss formulation uses the Markov assumption to depend on (X_t,Y) rather than the full state trajectory, alongside square integrability of the conditional density process.Square integrability is required to apply the martingale representation theorem and implies finite entropy under the stated condition.Source: §4 · pp. 3–4
• For each eligible terminal value and time horizon before T, the constructed control yields a weak controlled solution with the conditioned state-path law and uniquely minimizes energy among admissible Girsanov shifts.The result is stated on [0,r] for r < T.Source: §4 · p. 4
• The minimum-energy control lies in the range of the diffusion transpose because any component in ker σ leaves the state law unchanged while increasing energy.The resulting projection property allows simulation with available Brownian increments without numerical projection.Source: §4 · p. 4
• The control can be recovered from the induced state-drift shift using the Moore–Penrose inverse, even when the diffusion is noninjective.The missing Brownian component in the diffusion kernel is irrelevant to the minimum-energy control and controlled state equation.Source: §4 · pp. 4–5
• The local regression minimizer is a product-space conditional expectation, and its approximation converges in L² to the desired control as ε is used.The stated convergence is on [0,r] for r < T; the regression step is approximated in practice using a neural network.Source: §4 · p. 5

5.1 A Stochastic Control Problem

• The endpoint control uniquely minimizes a class of admissible one-sided Girsanov shifts, and its measure satisfies the stated conditional-law constraint.The informal stochastic-control result also states that V(r, y) = 0.Source: §5.1 · p. 7

5.2 Degenerate Nonlinear Diffusion

• For ε = 0.005, 0.01, and 0.1, separate learned controls trained with fixed-ε loss are compared with the numerical ground-truth Doob bridge using 15,000 independently simulated trajectories.Because t is autonomous, the ground truth can be computed from a one-dimensional backward Kolmogorov equation; Figure 1 shows sample paths, their empirical mean, and terminal target X(1).Source: §5.2 · pp. 7–8, Figure 1 · p. 8

5.3 Conditioned Image Generation

• For Fashion-MNIST inpainting, the model retains 648 of 784 image dimensions, keeping generated paths on an affine subspace without requiring an image-space density or score.Terminal images are projected onto retained principal components, and the bridge mixture has conditional marginals absolutely continuous with respect to the reference marginal before T.Source: §5.3 · pp. 8–9
• A small U-Net learns the unknown hidden-region control while the observed-region control is available analytically; generation combines predicted controls with Gaussian increments to produce multiple completions.Figure 2 presents eight held-out examples, with ground truth, condition, and generated image in successive columns.Source: §5.3 · p. 9, Figure 2 · p. 9

• Predictability and square integrability alone do not ensure that F is measurable with respect to the path of X, so minimum-entropy shifts must have the form σu.The representation relies on predictable representation for diffusion processes when the martingale problem is well posed; varying rank can limit geometric smoothness.Source: §6 · pp. 9–10
• The representation yields a nonsmooth analogue of the Doob h-transform and a method to approximate the minimum-entropy shift over Girsanov shifts.The paper connects this construction to causal optimal transport.Source: §6 · p. 10
• Setting u_t = ξ_t/Z_t(X) where Z_t(X) > 0, and defining it arbitrarily otherwise, yields the stated convergence in L2 up to every r < T.The integrand ξ exists and is unique under Assumption 2 and Theorem 4.Source: §6 · p. 10
• The constructed shift attains the minimum-entropy comparison, with equality only for the conditioned law and matching shift.The result is stated without requiring the density process Z to be strictly positive under dt ⊗ dP.Source: §6 · pp. 10–11
• The comparison admits pairs with a law absolutely continuous with respect to P, a predictable Brownian shift, and finite expected control energy.The admissible shift must also be F(X)-adapted.Source: §6 · p. 11

7 Conclusion

• The paper introduces regression losses for computing minimum-entropy adapted shifts when the diffusion is degenerate and the conditional law may be nonsmooth or singular.The construction uses the predictable representation property and requires neither a score function nor an invertible diffusion coefficient.Source: §7 · p. 11
• Under mild conditions, the resulting controls satisfy the minimum-entropy property within a class of admissible controls.Source: §7 · p. 11
• Extensions to manifold-valued and infinite-dimensional diffusions are identified as natural directions for future work.Source: §7 · p. 11

8 Impact Statement

• The work connects to diffusion models and shares the societal and ethical implications of generative modelling.Source: §8 · p. 11

A A Primer on Stochastic Analysis

• Weak existence and uniqueness in law under the conditioned measure suffice for the martingale representation theorem, including for SDEs without strong solutions.Source: A A Primer on Stochastic Analysis · p. 15

B A Discussion of the Tweedie Formulae

• For degenerate equations, the state-process filtration may be smaller than the Brownian filtration, so the orthogonal martingale term need not obviously vanish after conditioning.The representation result makes the Brownian integrand unique; its image under the diffusion coefficient gives the added state drift.Source: B A Discussion of the Tweedie Formulae · p. 15

C Proofs of the Theorems in Section 6

• Choosing the projection as a Borel function of the diffusion coefficient makes it usable under both the original and conditioned measures.The passage contrasts this with a measurable-selection choice that is not implementable in practice.Source: C Proofs of the Theorems in Section 6 · p. 16

C.1 Auxiliary Results

• The projection onto the diffusion coefficient's range can be predictable even when it is not continuous because the coefficient's rank varies.Predictability suffices for defining stochastic integrals.Source: §C.1 · p. 16

C.2 Proof of Theorem 4

• Using the raw state filtration’s predictable σ-field avoids relying on continuity of the projection process or switching between completed filtrations.The proof constructs a countable dense family of raw-predictable processes for the subsequent argument.Source: §C.2 · pp. 17, 21
• For μ0-almost every initial state x, (X, B) is a weak solution under Px with X0 = x.The proof first shows B remains Brownian under Px and verifies that the stochastic integral in the original SDE agrees with the Px-Itô integral.Source: §C.2 · pp. 20–21
• Disintegration over initial states and the fixed-initial representation theorem show that the orthogonal complement K⊥ is trivial.The proof uses the countable dense predictable family to transfer orthogonality to almost every conditioned fiber; it then proceeds to identify the closure of K.Source: §C.2 · pp. 21, 23
• The approximating processes converge to a predictable limit H that is invariant under Pt(X) almost everywhere, completing the asserted representation.The convergence is established using Itô’s isometry.Source: §C.2 · p. 23

C.3 Proof of Theorem 5

• The proof identifies eF with the conditional expectation of F given FT(X) by testing against square-integrable F(X)-predictable processes and using projection orthogonality.The predictable projection of ξ is projected again through Pt(X) to construct the F(X)-predictable integrand used in the argument.Source: §C.3 · pp. 24–25

C.4 Proof of Theorem 6

• Localization and Girsanov’s theorem establish the controlled dynamics, square integrability, and an exact energy identity on compact subintervals before extending the result to every r < T.The change of measure separates an initial density from the dynamic Girsanov shift.Source: §C.4 · pp. 25–27

C.5 Proof of Theorem 7

• The constructed controlled equation has the conditioned state-path law and attains the minimum-energy value, with uniqueness of the minimizing measure and control.The proof identifies the constructed measure with any other measure attaining the same value, then uses uniqueness of the semimartingale decomposition to identify the control.Source: §C.5 · pp. 28–29

D Statement and Proof of Theorem 3

• Theorem 10 considers absolutely continuous measures and predictable finite-energy shifts under which the driving process is Brownian.Source: D Statement and Proof of Theorem 3 · pp. 29–30
• Theorem 3 identifies uν,r as the unique minimum-energy Girsanov shift realizing νr.Source: D Statement and Proof of Theorem 3 · p. 30
• The pair (Qν,r, uν,r) attains zero, so V(r, νr) = 0.Source: D Statement and Proof of Theorem 3 · p. 31

E.1 Proof of Degenerate Tweedie formula-1

• The proof of Theorem 1 establishes the conditional-law and minimum-entropy claims on every interval [0, r], r < T.Source: §E.1 · p. 32

E.2 Proof of Degenerate Tweedie formula-2

• The proof of Degenerate Tweedie formula-2 applies Theorem 5 componentwise to Brownian increments and then uses the tower property.Source: §E.2 · p. 32

E.3 Proof of Degenerate Tweedie formula-3

• For Degenerate Tweedie formula-3, a conditional martingale identity is passed to the limit to prove equation 5.Source: §E.3 · pp. 32–33

F Terminal-Increment Regression for Additive Diffusions

• For additive diffusions with fixed noise, the terminal-increment target exactly replaces the local Brownian target and removes the auxiliary ε ↓ 0 limit.Source: F Terminal-Increment Regression for… · p. 33
• The regression target is the conditional expectation of Qt = (BT − Bt)/(T − t) given Xt and Y, and its unique minimizer is ū.Source: F Terminal-Increment Regression for… · pp. 33–34
• The path-adapted control admits this state-feedback version, and the terminal-increment loss recovers its controlled state law and minimum-energy property on [0, r].Source: F Terminal-Increment Regression for… · p. 34
• The additive model supplies its own state-feedback version without Assumption 3, whose controlled law and minimum-energy conclusions follow from Theorem 7.Source: F Terminal-Increment Regression for… · p. 34

G.1 Population regression

• Minimizing the squared population regression loss is equivalent to approximating the appropriate conditional expectation.Source: §G.1 · p. 35
• Under joint L2 convergence, both projected and unprojected losses recover the same minimum-energy control.Source: §G.1 · p. 35

G.2 Discretization of the projected target

• For projected targets, the displayed bound vanishes under ε ↓ 0 and h/ε → 0.Source: §G.2 · p. 36
• The projected-target discretization result assumes regular coefficients and a uniform lower bound on the smallest nonzero singular value of σ.Source: §G.2 · p. 36

G.3 The unprojected loss function

• The unprojected target is exact, so it has no stochastic-integral discretization error and needs no regularity assumption on Pt(X).Source: §G.3 · p. 38

G.4 Monte Carlo and optimization consistency

• Neural networks are proposed as a standard way to approximate the Tweedie-formula loss functions, with cited feasibility analysis stated to apply directly.Source: §G.4 · p. 38