CHUNG:2006 ; CHUNG:2016 , collision cascades KUBOTA:2007 ; BERLU:2008a ;
BERLU:2008b , evolution of damage and defect populations
resistivity:FLUSS:2004 ; magDefects:BACLET2002 ; JOMARD:2007 ;
magDefects:MCCALL:2006 ; ANISIMOV:2013 ; CONRADSOND:2014a ; KARAVAEV:2016 ;
deltaDefects:HERNANDEZ:2017 ; magDefects:HERNANDEZ2020 , including He
bubblesbubble:SCHWARTZ:2005 ; DREMOV:2008 ; JEFFRIES:2011 ; AO:2012 ;
KARAVAEV:2014 ; ROBINSON:2014 ; WU:2017 , changes in bulk properties
VALONE:2007 ; DREMOV:2013 ; DREMOV:2011 ; ARSENLIS:2005 ; LI:2011 ;
KARAVAEV:2017 ; puAging:Modulus:Exp , and the prospect of void
swellingALLEN:2015a ; ALLEN:2015b .

So far, there have been no experimental reports of void swelling in
$\delta$-Pu, despite having samples irradiated over decades and despite
calibrated attempts at accelerated aging in more recent years. It has been
proposed that the so-called net bias factor may be negligible in $\delta$ in
which case void swelling should be insignificant ALLEN:2015a . The bias is the
crux of the phenomenon:WOLFER:ASHKIN:SIPA:1975a ; WOLFER:ASHKIN:SIPA:1975b ;
SNIE  there would be no void growth and no swelling in the absence of biased
diffusion and defect segregation. However, this expectation is predicated on
models of elastic coupling between stress fields from the microstructure and
mobile defects using parameters taken from ab initio DFT calculations. It is
unclear if this conventional approach can be directly extended to as complex a
metal as Pu. Such uncertainty makes the behavior of other phases of Pu of
interest. The conventional theory (including the DFT-derived parameters) would
gain support if it can successfully explain swelling behavior in a diversity
of Pu phases. Arguably, finding discrepancies with experiment would be even
more informative.

In practice, the conventional theory explains both peak rates and trends in
void swelling in ordinary metals, but the approximations made in the analysis
and the uncertainties in the underlying parameters make the results as much
explanatory or extrapolative as predictive. Environmental parameters like
temperature, pressure, and radiation flux (or self-decay rates in Pu) may be
well-known for a given system. However, the microscopic response that
immediately follows radiation bombardment (effectively the average number of
atoms displaced and the average number of point defects ultimately left behind
by a single Pu-decay) is uncertain. Typically, reasonable parameter choices
can be made that quantitatively reproduce experimental swelling behavior in,
e.g., reactor structural materials. Theory-based modeling can then explain a
range of key swelling properties review:WOLFER:2000 , viz: 1) There is an
incubation period and transient initial delay during which swelling and void
growth is absent or greatly reduced. 2) Once the peak swelling rate (measured
against the rate of radiation-driven atomic displacements) is achieved it can
continue fairly steady-state even as the relative volume changes exceed 100%.
3) The peak swelling rate for a given material is also fairly independent of
temperatures and radiation fluxes over a wide range, as well as initial
microstructure. 4) In contrast, the incubation delay can vary greatly
depending on the starting microstructure, temperature, and radiation dose
rate.

That final point is understood from classical nucleation theory: under a
vacancy supersaturation, voids typically nucleate heterogeneously at pre-
existing helium bubbles, thus void swelling cannot commence until bubbles have
grown to the critical size SURH . A high concentration of defect traps like
small precipitates can delay the onset of void swelling by absorbing or
binding mobile defects like helium, vacancies, and interstitials. This leads
to many, smaller bubbles (delaying the first bubbles from reaching critical-
size), and it also facilitates the annihilation of intrinsic defects by
trapping both vacancies and interstitials together. However, such extrinsic,
microstructure-dependent delays mean that a null swelling result from a given
experiment does not preclude its appearance at later times. Indeed, the
microstructure is expected to change substantially during prolonged
irradiation. Under long exposures, each atom in a material can be displaced
from its lattice site on average 10 or 100 times, usually settling at a
completely new lattice site and only occasionally forming a point defect or
defect cluster. In light of this prospect, the theoretical intrinsic net bias
towards swelling may provide valuable insight into possible long-term void
swelling behavior SNIE .

In the theory, steady radiation bombardment generates quasi-stationary vacancy
and interstitial populations such that defect creation balances loss by mutual
annihilation or by absorption at microstructural sinks in the system. The
relevant microstructure includes network dislocations, dislocation loops,
cavities (high gas-density helium bubbles and low density voids),
precipitates, grain boundaries, etc. In many analyses, stochastic defect
production and spatially-dependent fluxes within a real microstructure are
approximated by mean field, steady-state diffusion calculations in idealized
geometries. The microstructure evolution may be comparably simplified:
cavities evolve according to mean field reaction rates while dislocations
multiply and annihilate according to simple rate equations for the aggregate
density SURH . Often the mean field problem is simplified further by assuming
isotropic diffusivity and by taking the biasing interactions to be the product
of relaxation volumes of mobile defects and hydrostatic stress fields of
microstructure sinks.

The computed average flux of vacancies and interstitials to each type of sink
is then summarized by separate bias factors, each a single dimensionless
number comparing the diffusive flux in the presence of the mutual interaction
and in the absence (i.e., unbiased diffusion). Their relative values determine
the overall tendency to vacancy-interstitial segregation and swelling. Network
dislocations contribute the most significant long-range elastic stresses, so
their bias factors dominate the situation. E.g., dislocations of density
$\rho$ (m-2) are modeled as an infinite straight edge dislocation in a
cylindrically-symmetric region with a 2-D Wigner-Seitz radius, where $\pi
R_{dis}^{2}=1/\rho$. Solving for the diffusion with and without an elastic
coupling that decays with distance as $1/r$, the dislocation bias factor is
approximated by:

$\displaystyle Z_{dis}(T,\Delta V_{R})$
$\displaystyle=ln\biggl{(}\frac{R_{dis}}{r_{dis}^{cor}}\biggr{)}\times$ (24)
$\displaystyle\bigg{[}\frac{K_{0}(r_{dis}^{cap}/R_{dis})}{I_{0}(r_{dis}^{cap}/R_{dis})}-\frac{K_{0}(r_{dis}^{cap}/r_{dis}^{cor})}{I_{0}(r_{dis}^{cap}/r_{dis}^{cor})}\bigg{]}^{-1}$

in terms of modified Bessel functions, where $\Delta V_{R}$ is the relaxation
volume (Eq. 11) of the mobile point defect, $R_{dis}$ is the 2D Wigner-Seitz
radius, $r_{dis}^{cor}=2b$ is the dislocation core radius (taken to be twice
the burgers vector, $b$), and
$r_{dis}^{cap}=\frac{(1+\nu)^{2}}{36\pi(1-\nu)}\frac{\mu|\Delta
V_{R}|}{k_{B}T}$ is the capture radius at which the point defect is absorbed
by the sink in terms of shear modulus, $\mu$, and Poisson ratio, $\nu$. Voids,
the other important class of sinks in this problem, have comparatively weak
stress fields and become increasingly unbiased sinks, $Z_{cav}\simeq 1$, in
the limit of large sizes.

The net difference at which interstitials and vacancies are absorbed then
determines whether dislocations climb and lattice sites are added while cavity
volume increases; this is expressed by the net bias factor. If the predicted
value is small, there should be little impetus for void swelling, incubation
periods can be prolonged, and any eventual swelling will be gradual. Assuming
that the overall total sink strength is dominated by the dislocation
contribution SNIE , the net bias factor becomes:

$B\simeq\frac{Z_{dis}^{int}}{Z_{dis}^{vac}}-\frac{Z_{cav}^{int}(x)}{Z_{cav}^{vac}(x)}$
(25)

as a function of cavity size, $x$. The most favorable conditions for void
growth are obtained when $Z_{cav}\rightarrow 1$, which will be assumed here.
The resulting material property, $B$, can be estimated from first principles
calculations.

However because of the assumed isotropy in the derivation of Eq. 25, the
result is only applicable to materials like cubic $\delta$-Pu. The site-
dependent defect energies for $\alpha$-Pu in Tabs. 9 and 10 already suggest
that diffusion will be highly anisotropic, potentially affecting the
calculations for net bias. The low crystal symmetry in $\alpha$ also implies
anisotropic lattice relaxation around defects, which requires coupling to the
full elastic stress tensor. Nevertheless, the relative V and SI relaxation
volumes are still primary quantities of interest here, and so we simply
compare relaxation volumes between $\alpha$\- and $\delta$-Pu as surrogates
for the relative tendency to void swelling. Based on Eq. 25 and the computed
defect relaxation volumes, swelling would be weak or nonexistent in
$\delta$-Pu but based on the defect volumes in $\alpha$-Pu it is significantly
likelier to be seen there.

####  V.1.1 $\delta$-Pu

Despite the simple crystal structure, there are significant uncertainties for
void swelling properties in $\delta$-Pu. Given the well-known errors for
equilibrium volumes of Pu phases, standard approaches like SP-GGA could have
difficulty with the defect relaxation volumes. Besides the usual elastic
interactions, recent calculations in (static mean field) SP-GGA also find a
magnetic component to Pu defects SHORIKOV:2013 ; spinDelta:HERNANDEZ:2019 ;
magDefects:HERNANDEZ2020 , which would contribute to the total interactions
and bias factors. There may furthermore be low energy electronic/magnetic
excitations of the defects with significant effects on the relaxation volume,
as considered in this work. Table 8 includes the SP-GGA results for a number
of metastable magnetic excitations in the 108-site $\delta$-Pu vacancy
supercell. The tabulated configurations represent a fraction of all possible
nearest-neighbor site combinations, but they are enough to show that magnetism
may be significantly coupled to lattice degrees of freedom. However as yet,
there is no evidence of a similar magnetoelastic effect for the interstitial
in $\delta$-Pu near the experimental density. This asymmetry may be intrinsic
or it may be an artifact of the external tension applied in the SP-GGA
calculations to approximate the experimental density. The simple vacancy spin-
flips in Table 8 are all disfavored in the more accurate SP+SO+OP-GGA
approximation. Non-collinear excitations may occur instead.

Independently of any particular DFT approximation, electronic fluctuations can
influence the defect volume whenever the energy for an atom to adjust its
electronic state (and f-shell size) is less than the energy stored in elastic
strain. I.e. such a localized excitation, $\Delta E$, could become favorable
if it alters an atomic volume by $\Delta V$, where the local pressure gives
${-P\Delta V}/{\Delta E}>1$. Some similar dimensionless parameter argument may
relate to the unusual bulk polymorphism and thermal expansion properties as
well. For example, thermally-accessible electronic excitations have been
suggested to explain the anomalous thermal expansion or Invar effect of
$\delta$-Pu.

It is notable that all of the vacancy energies listed in Table 8 are
thermally-accessible. A partition function for this limited state space can
provide expectation values for classical thermal fluctuations: the thermal
average vacancy relaxation volume ranges from -0.25 atomic volumes at 300K to
-0.29 at 600K. The results suggest that the net bias factor could be
temperature-dependent in $\delta$-Pu. However, the variation is only of order
10%, probably not enough to affect accelerated aging experiments. Some
temperature dependence is not unprecedented in void swelling; here it is a
magnetostructural version of the paraelastic effect already seen in some other
materials.

The large volume range for the $\delta$-Pu vacancy also implies that the
thermal-average relaxation volume will further depend on the local hydrostatic
stress. Similar so-called ‘diaelastic’ defect effects (also called modulus
interactions) are also seen in some systems. It means that the expected
vacancy relaxation volume will be weakly position dependent, affecting the
mean field diffusion analysis for the bias factors. For completeness, Table 6
includes estimates for the effective compressibilities of the ground-state
vacancy and self-interstitial species. In particular, the large effective
compressibility found for the vacancy implies that the modulus effects can be
quite significant in plutonium.

The different DFT approximations predict a considerable range of defect
parameters, but all of them imply that void swelling is weak or nonexistent in
$\delta$-Pu. The most extreme case is for the SP-GGA results at the
theoretical equilibrium; here the distorted monoclinic vacancy has a larger
absolute relaxation volume than the interstitial. In that case, the net bias,
$\Delta B$ is actually negative; no swelling is then expected as the vacancies
would be preferentially drawn to dislocations. If the SP+SO+OP-GGA results are
used, v=-0.9, i=0.85, and the net bias is effectively zero. The SP-GGA results
at -3GPa external stress depend on which distribution of magnetic vacancy
states is considered. For the putative vacancy ground state (with the smallest
relaxation volume), $\Delta B<0.2$; this net bias is comparable to that for
ferritic steels, which do not swell much SNIE . Thus, essentially no DFT
parameter set for $\delta$-Pu predicts much void swelling (i.e., DFT predicts
long incubation times and slow steady swelling, if any at all).

####  V.1.2 $\alpha$-Pu

We also examine defect formation energies and relaxation volumes in
$\alpha$-Pu, a more complicated structure with 8 crystallographically-distinct
sites. SP-GGA vacancy and interstitial calculations have been performed under
-5 GPa hydrostatic tensile stress for all of the sites. This is sufficient for
a preliminary comparison of void swelling tendencies with $\delta$-Pu.
Interstitial formation energies are consistently lower than vacancies. While
the lowest vacancy formation energy suggests an extremely high thermal
equilibrium density, the other striking result is that the relaxation volumes
are close to zero for almost all of the lattice sites, with large or small
formation energies alike. In contrast, the interstitial relaxation volumes are
all quite large. This would imply that $\alpha$-Pu is as prone to void
swelling as aluminum (one of the faster swelling materials known), although
the low symmetry and potentially anisotropic diffusion may affect this
prediction. The incubation delay will also depend on impurities and the
starting microstructure.

Remarkably, the two lowest energy sites are located in a layer-like
arrangement for both interstitials and vacancies. Intrinsically anisotropic
bulk diffusion in low-symmetry crystal structures may influence the mean field
diffusion fields to voids versus dislocations and so affect the bias
WOO:anisotropic:1988 ; BORODIN:1993 . Furthermore, if the diffusion approaches
quasi two-dimensionality it could invalidate the mean field approximations for
the bias factors. Conceivably, the spatial arrangement of surrounding sinks
(bubbles or voids and dislocations) could then become important.

The lower energy defect sites also coincide for vacancies and interstitials,
which may substantially increase their annihilation rates. In contrast, the
lowest energy sites for Ga are well-separated from the preferred vacancy
positions GaStable:SADIGH:2005 . This may inhibit Ga diffusion and prolong the
metastability of alloy-stabilized Pu. Substitutional He may conceivably show
similar behavior. In this light, a SP-GGA study of He defects is already
planned.

##  VI Conclusion

In this paper, we report extensive first-principles calculations of point
defect energies and structures in the $\alpha$\- and $\delta$-Pu phases. Two
broad surveys of point defect properties are reported in the SP-GGA
approximation along with a separate study using the SP+SO+OP-GGA
approximation, a method that predicts accurate bulk phase behavior. We discuss
the likelihood of void swelling for these two materials in light of the
calculated defect parameters. As might be expected from the equilibrium phase
diagram, defect behaviors are complex in this material, and this naturally
complicates any analysis of swelling behavior. However, conventional elastic
interactions will remain relevant to the theory even if other, more novel
mechanisms may contribute. The calculated material properties suggest that
void swelling is a greater possibility in $\alpha$-Pu. Void swelling is
expected to be weak or nonexistent in $\delta$-Pu, and indeed no void swelling
has been reported from experiment.

####  VI.0.1 Acknowledgements

We acknowledge Hector Lorenzana and Scott McCall for helpful discussions, and
for making us aware of their recent experimental progress by private
communication. We also thank A. Arsenlis, N. Barton, and J. Belof for project
support. This work was performed under the auspices of the U.S. Department of
Energy by Lawrence Livermore National Laboratory under Contract DE-
AC52-07NA27344.

*

###  .1 Appendix: Finite supercell-size errors

Our goal in this appendix is to study size-convergence of defect energies
under supercell volume/shape constraint as compared to under applied external
stress. For this purpose, consider a perfect crystal structure at pressure
$P_{0}$, temperature $T_{0}$, and non-hydrostatic stress state $\sigma_{0}$.
For brevity, we only consider here a single stress component, but
generalization to a full stress tensor is straightforward. As an example of a
crystal under a non-hydrostatic stress, let us consider an antiferromagnetic
fcc crystal, composed of ferromagnetic (001) layers, with the adjacent (001)
layers having opposite spin moments, i.e. L10 spin ordering. This is the
lowest-energy spin configuration for $\delta$-Pu within the SP-GGA
approximation, and happens to break the cubic symmetry. Hence, in general the
cubic fcc structure will be under a non-zero stress state that couples to
tetragonal shear distortion of the crystal leading to $c/a$-ratio that
deviates from unity. Let us denote by $\eta_{L}$ the tetragonal strain state
of the perfect crystal corresponding to the stress $\sigma_{0}$, and by
$V_{L}$ its volume at pressure $P_{0}$.

We now define the canonical energy as a function of volume and strain
$E(V,\eta)$, with the properties

$\displaystyle\frac{\partial E(V,\eta)}{\partial\eta}$ $\displaystyle=$
$\displaystyle\sigma.$ (26) $\displaystyle\frac{\partial E(V,\eta)}{\partial
V}$ $\displaystyle=$ $\displaystyle-P.$ (27)

These equations state the relevant stress-strain relationships. Before
proceeding, we define two relevant elastic constants

$\displaystyle\frac{\partial^{2}E(V,\eta)}{\partial\eta^{2}}$ $\displaystyle=$
$\displaystyle C^{\eta}.$ (28)
$\displaystyle\frac{\partial^{2}E(V,\eta)}{\partial V^{2}}$ $\displaystyle=$
$\displaystyle-\frac{K}{V}.$ (29)

Note that $K$ is the bulk modulus, and in the current example, $C^{\eta}$ is
the elastic constant for tetragonal shear.

We now proceed to define the enthalpy by a Legendre transformation of the
canonical energy

$\displaystyle H(P,\sigma)$ $\displaystyle=$ $\displaystyle
E(V(P,\sigma),\eta(P,\sigma))+P~{}V(P,\sigma)$ (30)
$\displaystyle-~{}\sigma~{}\eta(P,\sigma),$

where $\eta(P,\sigma)$ are $V(P,\sigma)$ are obtained by solving Eqs. 26 and
27. For the example of the perfect crystal described above, the different
variables in Eq. 30 can be defined as follows: $P\equiv P_{0}$,
$\sigma\equiv\sigma_{0}$, $V(P_{0},\sigma_{0})\equiv V_{L}$, and
$\eta(P_{0},\sigma_{0})\equiv\eta_{L}$.

Now consider the energetics of this crystal upon introduction of a
concentration $c_{0}$ of point defects. The energy and the enthalpy functions
can now be generalized to incorporate a finite point-defect concentration,
denoted by $\tilde{E}(V_{L},\eta_{L},c_{0})$, whenever system’s density and
strain are constrained, and $\tilde{H}(P_{0},\sigma_{0},c_{0})$, whenever the
external pressure and stress are controlled. Of course in practice, the defect
properties are calculated in periodic supercells at finite defect
concentration. Hence finite difference formulas are used to approximate
partial derivatives.

In the following, we consider two different procedures: (i) calculating
formation energy $\left.\frac{\partial\tilde{E}}{\partial
c}\right|_{V_{L},\eta_{L},c_{0}}$ of point defects allowing for no relaxation
of either cell shape or volume, and (ii) calculating formation enthalpy
$\left.\frac{\partial\tilde{H}}{\partial c}\right|_{P_{0},\sigma_{0},c_{0}}$
of point defects allowing for relaxations of both supercell shape and volume.
In this case, we can also define a relaxation volume as in Eq. 10, and
analogously a relaxation strain

$\Delta\eta_{R}=\frac{\partial\tilde{H}}{\partial c}.$ (31)

Let us then start with case (i). In this case, the defect supercell volume
$V_{D}$ as well as its strain state $\eta_{D}$ are kept fixed and equal to the
perfect lattice $V_{L}$ and $\eta_{L}$, i.e. $V_{D}=V_{L}$, and
$\eta_{D}=\eta_{L}$. The finite-size error of the formation energy calculated
in a defect supercell containing N sites can be expanded in a Taylor serier,
whose first term is

$\tilde{E}(V_{L},\eta_{L},1/N)-\tilde{E}(V_{L},\eta_{L},0)-\Delta
E_{F}=\frac{c_{0}}{2}\frac{\partial^{2}\tilde{E}}{\partial c^{2}}+....$ (32)

Note that $\Delta E_{F}=\partial\tilde{E}/\partial c$.

Now let us consider case (ii). In this case, the defect supercell volume
$V_{D}$ and the tetragonality parameter $\eta_{D}$ are allowed to relax in
order to maintain the pressure $P_{0}$ and the stress $\sigma_{0}$. Hence, the
finite-size error of the formation enthalpy, calculated in a defect supercell
containing N sites can be expanded in a Taylor serier, whose first term is

$\displaystyle\tilde{H}(P_{0},\sigma_{0},1/N)-\tilde{H}(P_{0},\sigma_{0},0)-\Delta
H_{F}=$ $\displaystyle\frac{c_{0}}{2}\frac{\partial^{2}\tilde{E}}{\partial
c^{2}}-\frac{K_{L}}{V_{L}}~{}\Delta
V_{R}^{2}-C_{L}^{\eta}~{}\Delta\eta_{R}^{2}+...,$ (33)

where $K_{L}$ and $C_{L}^{\eta}$ are the bulk modulus and elastic constant of
the perfect crystal. The above result is easily obtained by Taylor expansion
of the generalized enthalpy $\tilde{H}(P_{0},\sigma_{0},1/N)$ about zero
defect concentration, using the stress-strain relations Eqs. 26 and 27, as
well as the definitions of the elastic constants Eqs. 28 and 29. In this way,
it can also be shown that $\Delta H_{F}=\Delta E_{F}$.

Equation .1 confirms that finite supercell-size errors introduced in
calculated defect properties can often be reduced significantly whenever
supecell shape and volume relaxations are allowed. The above derivation relies
on the defect concentration be small enough, or supercell sizes large enough,
such that corrections due to periodic image interactions can be expanded in
low-order perturbation theory. This condition is usually satisfied in defect
supercells exceeding 100 atoms. However, it should be noted that even in large
supercells, shape and volume relaxations cannot fully account for all the
periodic image interactions. They can be considered the lowest order terms in
a multipole expansion of these interactions.

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# Informed Correctors for Discrete Diffusion Models

Yixiu Zhao
Stanford University
<EMAIL_ADDRESS>
&Jiaxin Shi
Google DeepMind
<EMAIL_ADDRESS>
Lester Mackey
Microsoft Research New England
<EMAIL_ADDRESS>
&Scott Linderman
Stanford University
<EMAIL_ADDRESS>
###### Abstract

Discrete diffusion modeling is a promising framework for modeling and
generating data in discrete spaces. To sample from these models, different
strategies present trade-offs between computation and sample quality. A
predominant sampling strategy is predictor-corrector $\tau$-leaping, which
simulates the continuous time generative process with discretized predictor
steps and counteracts the accumulation of discretization error via corrector
steps. However, for absorbing state diffusion, an important class of discrete
diffusion models, the standard forward-backward corrector can be ineffective
in fixing such errors, resulting in subpar sample quality. To remedy this
problem, we propose a family of informed correctors that more reliably
counteracts discretization error by leveraging information learned by the
model. For further efficiency gains, we also propose $k$-Gillespie’s, a
sampling algorithm that better utilizes each model evaluation, while still
enjoying the speed and flexibility of $\tau$-leaping. Across several real and
synthetic datasets, we show that $k$-Gillespie’s with informed correctors
reliably produces higher quality samples at lower computational cost.

##  1 Introduction

Denoising diffusion models are powerful generative models for high-dimensional
data [1, 2, 3]. The central idea of diffusion models is to gradually corrupt
data into noise using a “noising” or “forward” process and then to train a
parameterized model (usually a deep neural network) to learn its time
reversal, commonly known as the “denoising” or simply the “backward” process.
In continuous domains such as image generation, the forward process is usually
defined by gradual injection of Gaussian noise and scaling, transforming the
data distribution close to a standard normal. The backward process is then
approximated by learning the gradient of the log density (also known as the
“score function”) of the marginal distribution. Samples can be drawn from a
trained model by first generating Gaussian noise and then simulating the
backward process using the score information. Diffusion generative models are
currently the dominant framework for image generation and can generate high-
resolution images with stunning details and style given prompts by the users
[4, 5].

Given the success of diffusion models in continuous domains, recent work has
explored diffusion modeling in discrete domains [6, 7, 8, 9, 10, 11, 12, 13],
with applications to language [13], protein sequences [14, 15], graphs [16],
and more. Notably, Campbell et al. [9] developed TauLDR, a general framework
for discrete denoising diffusion in continuous time. They formulated the
forward and backward processes as continuous time Markov chains (CTMCs), and
their model learns to predict the distribution over denoised data via an
evidence lower bound (ELBO) objective. Lou et al. [13] adopted a very similar
strategy but instead directly parameterize the concrete score function — an
analog of the continuous score function in discrete spaces.

Despite the conceptual progress on discrete diffusion, many challenges in
implementation still limit the effectiveness of these models. One major
challenge lies in the trade-off between efficiency and accuracy in simulating
the continuous-time backward process. Song et al. [17] proposed using
approximate MCMC correctors to correct for the discretization error in
simulating diffusion backward process. Campbell et al. [9] extended predictor-
corrector sampling to discrete diffusion models and discovered that the
corrector steps significantly impact generative performance. However, a clear
understanding of how the choice of corrector steps influences sample quality
is still lacking.

Can further gains be made from leveraging different sampling methods? We give
an affirmative answer. By analyzing the forward-backward corrector used in
TauLDR, we identify failure modes in absorbing state diffusion and develop a
simple synthetic “countdown” dataset that exposes the pathologies of standard
corrector steps. To address this shortcoming:

  * •

We propose a family of “informed correctors” and accompanying training schemes
that further capitalize on the score information gathered by the model during
training. We use them to guide the sampling process, correcting mistakes and
accelerating mixing.

  * •

We propose an alternative to $\tau$-leaping called $k$-Gillespie’s, which
takes a fixed number of updates $k\geq 1$ per backward step. We show that it
generally achieves better sample quality than $\tau$-leaping for the same
number of model evaluations.

We compare these models on the countdown dataset as well as benchmarks for
discrete sequence generation including gene expression the monophonic music
sequences, We show that informed correctors, together with our other
algorithmic and architectural proposals, produce higher quality samples with
fewer model evaluations.

##  2 Background

###  2.1 Continuous-Time Discrete Diffusion Models

Consider a data point $x_{0}\in\mathcal{S}$ which is assumed to be sampled
from the data distribution $p_{\text{data}}(x_{0})$. Denoising diffusion
models [1, 2, 3] seek to model this unknown data distribution by progressively
adding noise to data samples until their distribution converges to a simple
target, such as a uniform distribution. Then, a deep neural network is trained
to invert this process and remove the noise. This reverse process forms a
generative model from which new samples can be drawn. The noising and
generative processes are also called the _forward_ and backward processes,
respectively. We will adopt this convention for the rest of the paper.

Continuous-time diffusion models define a forward process with noising
distribution $q_{t}(x_{t}|x_{0})$ and marginals $q_{t}(x_{t})=\int
q_{t}(x_{t}|x_{0})q_{0}(x_{0})\mathrm{d}x_{0}$, with $q_{0}=p_{\text{data}}$
and $q_{T}\approx\pi$ where $\pi$ is the limiting distribution of the forward
process, which is assumed to be a simple distribution. Then, a parameterized
process with marginals $p^{\theta}_{t}$ is learned to match the true backward
process.

When the data of interest is discrete in nature (e.g., text, protein
sequences, neural spike counts), the forward-process can be described with a
continuous-time Markov chain (CTMC), which is captured by the initial
distribution $q_{0}$ and a transition rate matrix $R_{t}\in\mathbb{R}^{S\times
S}$, where $S=|\mathcal{S}|$ denotes the cardinality of the data space. The
off-diagonal terms $R_{t}(x,y),x\neq y$ of the matrix represent the rate of
state $x$ transitioning to state $y$, and the diagonals $R_{t}(x,x)$ are
defined as the negative sum of all other elements in the same row:
$R_{t}(x,x)=-\sum_{y\neq x}R_{t}(x,y)$. Under the assumption that $R_{s}$ and
$R_{t}$ commute for all $s$ and $t$, the finite-time transition probabilities
satisfy

$q_{t+\Delta t\mid t}(y\mid x)=\left[\exp\left\\{\int_{t}^{t+\Delta
t}R_{s}\,\mathrm{d}s\right\\}\right](x,y)=I(x,y)+R_{t}(x,y)\Delta t+o(\Delta
t),$ (1)

where $I$ represents the identity matrix, and in the square brackets we
compute a matrix exponential. It can then be shown [9] that the time reversal
of the above CTMC is another CTMC with the transition rate

$\tilde{R}_{t}(y,x)=R_{t}(x,y)\frac{q_{t}(x)}{q_{t}(y)}=R_{t}(x,y)\sum_{x_{0}}\frac{q_{t|0}(x\mid
x_{0})}{q_{t|0}(y\mid x_{0})}q_{0|t}(x_{0}\mid y).$ (2)

Campbell et al. [9] approximate the backward rate by learning a parameterized
denoising model $p^{\theta}_{0|t}(x_{0}\mid y)\approx q_{0|t}(x_{0}\mid y)$.
Alternatively, Meng et al. [18] and Lou et al. [13] observed that the ratio
$s_{t}(y)_{x}\triangleq\nicefrac{{q_{t}(x)}}{{q_{t}(y)}}$ plays a similar role
as the continuous score function $\nabla\log q_{t}(y)$, up to an additive
constant. Therefore, they proposed to learn an approximate function,
$s^{\theta}_{t}(y)$. We adopt the denoising parameterization of Campbell et
al. [9] in our work, but use both of these objects in our exposition, keeping
in mind the relationship between them:

$s^{\theta}_{t}(y)_{x}=\sum_{x_{0}}\frac{q_{t|0}(x\mid x_{0})}{q_{t|0}(y\mid
x_{0})}p^{\theta}_{0|t}(x_{0}\mid y).$ (3)

Regardless of the parameterization, the training objective can be given by the
variational lower bound of the data log likelihood in terms of the
parameterized backward rate
$\tilde{R}_{t}^{\theta}(y,x)=R_{t}(x,y)s^{\theta}_{t}(y)_{x}$:

$-\log p(x_{0})\leq\mathcal{L}(\theta)=\int\mathbb{E}_{q_{t|0}(y\mid
x_{0})}\Big{[}\sum_{x\neq
y}\big{\\{}\tilde{R}_{t}^{\theta}(y,x)-R_{t}(x,y)\frac{q_{t|0}(x\mid
x_{0})}{q_{t|0}(y\mid
x_{0})}\log(\tilde{R}_{t}^{\theta}(y,x))\big{\\}}\Big{]}\,\mathrm{d}t+C,$ (4)

where $C$ is a constant independent of $\theta$. This is the standard evidence
lower bound (ELBO) objective for training continuous time discrete diffusion
models.

Note that this objective is superficially different from the one in Campbell
et al. [9], which practically requires two Monte Carlo samples to estimate
instead of one. However, as Shi et al. [19] have noted, these objectives are
mathematically equivalent, therefore we present the simpler form here. We
provide a derivation of this fact in Appendix A and refer readers to Shi et
al. [19] for more detail. In practice, we choose the objective that matches
existing work for fair comparison.

###  2.2 Sequence Modeling with the Absorbing Forward Process

For real-world applications, the data is often a vector or sequence $x^{1:D}$
of length $D$, where each dimension of the data $x^{d}$ belongs to a
vocabulary, and the full state space is the product of the vocabularies and
thus exponentially large. With a shift in notation, we now use
$\mathcal{S}=\\{1,\dots,S\\}$ to denote the “vocabulary” set instead, whereas
the full space is represented as $\mathcal{S}^{D}$.

Given the factorization of this space, it is natural to model the forward
process as a composition of identical independent processes on each dimension,
where the rate over the entire state $R_{t}$ can be written as

$R_{t}(x,y)=\beta(t)\sum_{d}R_{b}(x^{d},y^{d})\mathbbm{1}_{\\{x^{\setminus
d}=y^{\setminus d}\\}}$ (5)

for a base rate matrix $R_{b}\in\mathbb{R}^{S\times S}$, absorbing the time
dependence into the scalar coefficient $\beta(t)$. With this factorization of
the forward rate, the backward rates are also factorized, but now with a
different time-dependent rate $\hat{R}^{d}_{t}$ for each dimension $d$. This
means that the objective in 4 can be decomposed nicely into a sum over
dimensions, as detailed by Campbell et al. [9].

For generic discrete data, two forward processes are most commonly used: the
uniform process and the absorbing state process. We focus on the absorbing
state process in this paper, for which the base rate matrix for each dimension
can be written as

$R^{\text{absorb}}_{b}(x,y)=\begin{cases}1&y=\textsc{Mask},\;x\neq\textsc{Mask}\\\
-1&y=x,\;x\neq\textsc{Mask}\\\ 0&\text{otherwise},\end{cases}$ (6)

where we augment the vocabulary set $\mathcal{S}$ by introducing the Mask
token, which represents the absorbing state. The absorbing state diffusion is
important to consider because it is very commonly used to model sequence data
without ordinal structure (e.g., text and protein sequences). It is also
intimately connected to masked language models [8, 20, 21].

We use special notation for the absorbing state process. For a sequence
$x\in\\{\textsc{Mask},1,\dots,S\\}^{D}$, we use the mask operator $y=M^{d}(x)$
to denote the sequence obtained by changing the $d$th component of $x$ to
Mask. Finally, we denote the masked indices of $x$ by
$\mathcal{M}(x)\equiv\\{d:x^{d}=\textsc{Mask}\\}$. Then the set of non-mask
indices is denoted by the set complement $\overline{\mathcal{M}}(x)$.

###  2.3 Sampling From the Backward Process

#### Gillespie’s method

Given learned rates $\tilde{R}_{t}^{\theta}$ for the backward process, one can
approximately sample from the backward Markov jump process using Gillespie’s
algorithm [22] which computes holding times at the current state, and
recomputes the backward rates after the holding time has passed and a state
change takes place. The approximation arises when the backward rates are
treated as constant over short time intervals, when they in fact change
continuously in time. Gillespie’s algorithm changes a single component of the
state at a time, which is computationally expensive for generating high-
dimensional data.

#### Tau leaping

Instead, Campbell et al. [9] proposed to use $\tau$-leaping, an approximate
sampling method that allows simultaneous updates to multiple dimensions for a
single model evaluation. $\tau$-leaping assumes that the backward process
transition rate is constant between $[t-\tau,t)$, regardless of whether states
change during that interval. Under this assumption, the number of transitions
in the interval is a Poisson random variable, and several transition events
can be applied simultaneously.

###  2.4 Corrector Steps for $\tau$-leaping

Because of the approximate nature of the $\tau$-leaping steps, this approach
accumulates simulation errors over the course of the backward process. To
combat this problem, additional corrector steps can be used to make sure that
the marginal distribution of samples $x_{t}$ at time $t$ matches $q_{t}$.
Campbell et al. [9] show that a corrector process with rates:

$R^{c}_{t}(x,y)=R_{t}(x,y)+\tilde{R}_{t}(x,y)$ (7)

leaves the stationary distribution $q_{t}$ invariant. In practice, one must
use the learned backward rates $\tilde{R}_{t}^{\theta}$ as an approximation
for the true backward rates $\tilde{R}_{t}$. We refer to this as the forward-
backward corrector, as it is simply a summation of the forward and backward
rates. Campbell et al. [9] demonstrate that the forward-backward corrector
increases sample quality in practice.

#### Correcting for the absorbing diffusion

The necessity of corrector steps is particularly apparent when $\tau$-leaping
is used in conjunction with an absorbing forward process. In the reversal of
the absorbing process, only transitions from the absorbing state to other
states are allowed, which is reflected by the structure of $\tilde{R}_{t}$.
Thus, once a token has been generated, it cannot be erased or changed,
rendering it impossible for the predictor to fix its own errors. Forward-
backward corrector steps mitigate this issue by introducing the forward term
$R_{t}$, allowing transitions in the forward direction, resulting in generated
tokens being “masked out” again with uniform probability.

Though forward-backward corrector steps solve the issue in theory, we note
that the forward-backward corrector is far from optimal in practice. It can
only correct mistakes by masking them at random. Ideally, we would like the
corrector to make informed transitions based on which states are more likely
under the model. This observation motivates us to find a better alternative to
the forward-backward corrector.

##  3 Methods

This section presents our main contributions. Focusing on discrete diffusion
models with an absorbing forward process, we propose a modification of
Gillespie’s algorithm, a classic algorithm for simulating CTMCs, to
approximate the backward process with fewer model evaluations. Then, motivated
by the discussion above, we develop novel corrector steps for the reverse
process. Taking inspiration from the Stein operator literature, we present
informed correctors, a family of Markov transition operators that leverage
score information learned by the model to more efficiently address mistakes in
the reverse process and improve sample quality.

###  3.1 $k$-Gillespie’s Algorithm

In the absorbing state reverse process, a change in the state happens if and
only if a new token is generated from the absorbing state. This means that the
number of state changes is exactly equal to the number of positions $D$,
which, in Gillespie’s algorithm, also corresponds to $D$ model evaluations.

Unlike Gillespie’s, $\tau$-leaping can freely choose the number of model
evaluations by changing the step size $\tau$. This enables a trade-off between
sample quality and computation. Unfortunately, a Poisson-distributed number of
updates in the $\tau$-leaping algorithm means that on some steps, no updates
are made at all. One model evaluation can lead to multiple coordinates being
updated at once — a loss of accuracy — or no coordinates being updated at all
— a loss of efficiency.

To address this issue with $\tau$-leaping, we propose a simple modification of
Gillespie’s algorithm. A single model evaluation provides information about
transition rates for all dimensions $D$. Rather than updating only the
dimension with the smallest holding time, we update the $k\geq 1$ dimensions
with the smallest holding times simultaneously. By approximating the rate as
constant within the time of these $k$ transitions, this allows us to generate
samples using only $\lfloor\frac{D}{k}\rfloor$ model evaluations. Similar to
$\tau$-leaping, we can also offset the accumulation of errors due to this
approximation by applying corrector steps after each set of updates. We
provide pseudo-code for this in Algorithm 1 in the appendix.

###  3.2 Informed Correctors

We propose the following family of correctors for discrete diffusion sampling
by combining the symmetrized forward process and with an additional “corrector
operator,” $A_{t}$,

$R^{c}_{t}(x,y)=\begin{cases}(R_{t}(x,y)+R_{t}(y,x))\cdot A_{t}(x,y)&x\neq
y\\\ -\sum_{z}R^{c}_{t}(x,z)&x=y\end{cases}$ (8)

Here, $A_{t}$ is the matrix representation of a Markov transition operator
that leaves the marginal distribution $q_{t}$ invariant. That is, a continuous
time Markov jump process generated by $A_{t}$ has stationary distribution
$q_{t}$. One way to obtain a corrector operator is by appealing to the
literature on Stein operators. For example, a Zanella Stein operator [23] is
of the form,

$A_{t}(x,y)=\begin{cases}\kappa(\frac{q_{t}(y)}{q_{t}(x)})&y\neq x\\\
-\sum_{z}A_{t}(x,z)&y=x\end{cases},$ (9)

where $\kappa$ is a positive function satisfying
$\kappa(t)=t\kappa(1/t)$.111Zanella [23] defines a neighborhood of state $x$,
but premultiplying by the symmetrized forward rate matrix
$R_{t}(x,y)+R_{t}(y,x)$ achieve a similar effect, allowing us to simplify the
presentation of $A_{t}$. In this work, we will focus on the minimum
probability flow (MPF) Stein operator [24, 25], where $\kappa(t)=\sqrt{t}$,
and the Barker Stein operator [26, 27], where $\kappa(t)=\frac{t}{t+1}$, two
examples of Zanella Stein operators that have been found to work well in
practice [26].

In practice, we would like to use $A^{\theta}_{t}$ to approximate $A_{t}$,
where $A^{\theta}_{t}$ uses the learned concrete score $s^{\theta}_{t}$ in
place of the true $s_{t}$. However, it is important to note that this
approximation is not always viable, since the score information
$\frac{q_{t}(x)}{q_{t}(y)}$ is not learned by the model for all pairs of
$(x,y)$, but only for those that are allowed by the forward process. This
follows from the ELBO objective in 4, where the learning signal is weighted by
the forward rates. Therefore, the model can learn poor approximations for
$\frac{q_{t}(x)}{q_{t}(y)}$ if the forward rate $R_{t}(x,y)$ is $0$ or very
small.

While diffusion with a uniform forward process is not affected by virtue of
sampling transitions between all pairs of states during training, this
limitation becomes apparent for absorbing state diffusion, which only allows
mask to non-mask transitions but not vice versa. In the next section, we show
why having informed correctors is especially important in the absorbing case
and illustrate ways to learn informed correctors with small adjustments to the
score function architecture.

##  4 Learning Informed Correctors for Absorbing State Diffusion

We develop an approach for learning informed correctors for discrete diffusion
models with absorbing forward processes. Specifically, we modify the
parameterization and architecture such that when trained with the _same_ ELBO
objective, the model learns the necessary score information efficiently. Note
that this change in architecture is only required for the absorbing state
diffusion, and for uniform process diffusion informed correctors can be
applied directly without changing the learning scheme.

###  4.1 Alternative Parameterization for Efficient Score Evaluation

The corrector rates in 8 and 9 provide a nonzero rate of converting the token
at the $d$-th dimension of $x$ back into a mask. These rates depend on the
score function
$s_{t}(x)_{M^{d}(x)}\triangleq\nicefrac{{q_{t}(M^{d}(x))}}{{q_{t}(x)}}$.
However, $s_{t}^{\theta}(x)_{M^{d}(x)}$ is not learned because
$R_{t}(M^{d}(x),x)=0$ which zeros out the reverse rate
$\tilde{R}_{t}^{\theta}$ in 4. We can circumvent this issue by inverting the
the learned score $s_{t}^{\theta}(M^{d}(x))_{x}$ to get the desired ratio.
Nevertheless, this approach is very computationally inefficient: to compute
the transition rates for all non-mask dimensions, we need to evaluate our
model on $|\overline{\mathcal{M}}(x)|$ different inputs.

We address this challenge via an alternative parameterization of the denoising
model. Let $f^{\theta}(x,t)$ represent a neural network that implements an
approximate denoising distribution. It takes in an observation $x$ and time
$t$ and outputs a tensor in $\mathbb{R}^{D\times S}$. Let
$f^{\theta}(x,t)_{d,i}$ denote the $(d,i)$-th output of the neural network,
which represents the denoising probability $p^{\theta}_{0|t}(x^{d}_{0}=i\mid
x)\triangleq f^{\theta}(x,t)_{d,i}$.

For an absorbing process, observe that for $d\in\overline{\mathcal{M}}(x)$ we
know that $q_{0|t}(x^{d}_{0}=i\mid x)=\mathbbm{1}_{\\{x^{d}=i\\}}$, which
implies that the entire output probability vector $f^{\theta}(x,t)_{d,:}$ is
unused. We propose to use these output dimensions to approximate
$q_{0|t}(x^{d}_{0}\mid M^{d}(x))$ instead. With that information, we can
compute the required score information via,

$s_{t}(x)_{M^{d}(x)}=\left(\frac{q_{t}(x)}{q_{t}(M^{d}(x))}\right)^{-1}\approx\left[\sum_{i=1}^{S}\frac{q_{t|0}(x\mid
x_{0}=i)}{q_{t|0}(M^{d}(x)\mid x_{0}=i)}f^{\theta}(x,t)_{d,i}\right]^{-1}.$
(10)

To summarize, for $d\in\mathcal{M}(x)$, the network outputs the same denoising
distribution as before; for $d\in\overline{\mathcal{M}}(x)$, the network
operates as if the input has a mask token at dimension $d$. With this
parameterization, a single function evaluation is sufficient for computing the
informed corrector rates for all $D$ dimensions.

###  4.2 Learning the Full Score Information Using Hollow Transformers

Despite the conceptual simplicity, for this parameterization to work in
practice, one must make sure that for $d\in\overline{\mathcal{M}}(x)$
information about $x^{d}$ does not leak to the output $f^{\theta}(x,t)_{d,i}$.
To achieve this, we use a hollow transformer [28], a two-stream self-attention
architecture where each stream runs causal attention in a different direction.
The embeddings are then offset by one position so that the embedding at
position $d$ does not attend to the corresponding input position. By combining
these two directional streams, the hollow transformer satisfies the property
that, for continuous $x$,

$\frac{\partial}{\partial x^{d}}f^{\theta}_{HT}(x,t)_{d,i}=0\quad\forall
i\in\\{1,\dots,S\\}.$ (11)

In other words, the network output at dimension $d$ is agnostic to the input
at the same dimension $d$. We provide more details of the model architecture
in Section 6.

#### Learning the Hollow Transformer with the ELBO Objective

The parameterization in 10 can be learned with the hollow transformer without
changing the ELBO objective 4. To see why, consider a pair of observations $x$
and $x^{\prime}=M^{d}(x)$ such that $x^{d}\neq\textsc{Mask}$. By the property
of the hollow transformer,
$f^{\theta}_{HT}(x,t)_{d,i}=f^{\theta}_{HT}(x^{\prime},t)_{d,i}$. However,
since the forward rate $R_{t}(\cdot,x)$ is 0 under the absorbing forward
process, the backward rate $\tilde{R}_{t}^{\theta}(x,\cdot)$ is also 0. This
means that during training, gradients only flow through
$\tilde{R}_{t}^{\theta}(\mathcal{M}^{d},\cdot)$, which encourages the network
output $f^{\theta}_{HT}(x,t)_{d,i}$ to approach the true denoising
distribution $q^{d}_{0|t}(x_{0}^{d}=i\mid M^{d}(x))$ which is conditioned on
the masked state $M^{d}(x)=x^{\prime}$. In other words, since the model does
not see $x^{d}$ when producing the output at dimension $d$, it always tries to
produce outputs as if $x^{d}=\textsc{Mask}$ to minimize the loss.

##  5 Related Work

#### Discrete diffusion models and predictor-corrector sampling.

Sohl-Dickstein et al. [1] derived a discrete diffusion model but only for
binomial variables. Hoogeboom et al. [6], Song et al. [7] extended the
formulation to categorical variables using a uniform noising distribution,
which was further generalized by Austin et al. [8] to enable any corruption
processes specified with a transition matrix. Austin et al. [8] also discussed
the continuous-time limit of the proposed discrete diffusion model. A
continuous-time objective amenable to training was first proposed by Campbell
et al. [9] and further developed by Benton et al. [12], Lou et al. [13], Shi
et al. [19]. In addition to this line of work on formally defining the forward
and backward diffusion process and deriving likelihood-based training
objectives, there is also a family of diffusion-inspired models targeted at
discrete data, such as SUNDAE [29], MaskGIT [30], and SDDM [28]. These models
are typically trained by an objective that does not necessarily correspond to
maximizing the likelihood of data, such as a (weighted) denoising loss (i.e.,
cross-entropy for discrete data) or a generalization of ratio-matching losses
[28]. The first use of predictor-corrector samplers appeared in Song et al.
[17] for continuous diffusion models. Campbell et al. [9] used correctors to
improve their $\tau$-leaping samplers for discrete diffusion. Relatedly,
Lezama et al. [31] also proposed a corrector scheme to improve MaskGIT
sampling.

#### MCMC in discrete spaces.

The design of our corrector scheme involves finding continuous-time Markov
chains that keep our current-time marginals as the stationary distribution.
This problem has largely been studied for designing MCMC algorithms in
discrete spaces. The Barker and MPF correctors are inspired by recent work
[26, 32] that uses Zanella’s informed MCMC kernels [23] to construct Stein
operators or transition rate matrices that have a target stationary
distribution. In addition to the locally balanced proposals studied in Zanella
[23], there are several other discrete MCMC kernels relying on continuous
relaxation and combining ideas from Gibbs or Langevin samplers [33, 34, 35]
which would be interesting to explore as correctors for discrete diffusion
models in future work.

##  6 Experiments

We evaluate k-Gillespie’s algorithm and informed correctors on synthetic and
real datasets for both unconditional and conditional sampling tasks. We train
a hollow transformer architecture augmented for denoising diffusion using
standard implementation techniques [17]. The inputs to the transformer are
treated as one-hot variables, destroying all ordinal structure of the data.
Our networks use 5M to 8M parameters depending on the task. Training is done
using a single A100 GPU with 80GB of memory on an academic cluster.

To study efficiency-quality trade-offs, we measure different dataset-dependent
metrics and also record the number of function evaluations (NFE) needed to
produce each sample under different sampling schemes. For $\tau$-leaping, we
control NFE via tuning the $\tau$ parameter, which is the inverse of the
number of predictor steps. For our $k$-Gillespie’s, we change $k$ from $1$ to
$3$, with higher values of $k$ resulting in fewer NFEs. Keep in mind that each
corrector step also requires one function evaluation.

###  6.1 The Countdown Dataset

To check our intuition that standard forward-backward correctors would perform
poorly for absorbing diffusion processes (see Section 2.4), we create a
synthetic sequence dataset with strong position-wise correlation structure and
test generation quality for different corrector steps. Given a state size $S$
and sequence length $D$, a data sequence $x^{1:D}$ is drawn according to the
following rule:

$x^{1}\sim\text{Uniform}\\{1,\dots,S\\},\quad x^{d+1}\mid
x^{d}\sim\begin{cases}\delta_{x^{d}-1}&x^{d}\neq 0\\\
\text{Uniform}\\{1,\dots,S\\}&x^{d}=0\end{cases}.$ (12)

We evaluate sample quality for this dataset using the error rate measured by
the proportion of generated tokens that fail to count downward from the
previous token:

$r_{err}(\\{\hat{x}_{n}\\}_{1\dots
N})=\frac{1}{ND}\sum^{N}_{n=1}\sum^{D-1}_{d=1}\mathbbm{1}_{\\{\hat{x}_{n}^{d+1}\neq\hat{x}_{n}^{d}-1\\}}\mathbbm{1}_{\\{\hat{x}_{n}^{d}\neq
0\\}}.$ (13)

The results are shown in Figure 1. We confirm the observation from [9] that
more corrector steps (thicker lines) improves sample quality. However, the
forward-backward corrector (orange curves) notably benefits the least from
increased corrector steps in proportion of errors. This confirms the intuition
that the forward-backward corrector is bad at correcting mistakes due to
masking tokens uniformly at random.

Except for the MPF corrector on the proportion of errors metric,
$k$-Gillespie’s (solid curves) greatly outperforms $\tau$-leaping across
different setting, achieving better sample quality with fewer than half as
many model evaluations. Notably, MPF (pink curves) outperforms other
correctors by an order of magnitude in reducing errors. We demonstrate this
visually in Figure 2.

Figure 1: Comparison between different sampling schemes on the countdown
dataset. The number of function evaluations is tuned by changing $k$ and
$\tau$. Each run generates 500 sample sequences and standard deviations are
computed from 5 runs with different random seeds. Figure 2:  Samples taken
using uninformed and informed correctors on the countdown dataset. The top row
shows a typical data sequence. The middle sequence is sampled using
$k$-Gillespie’s and the MPF corrector (NFE = 163), and the bottom sequence is
sampled with $\tau$-leaping and the forward-backward corrector (NFE = 190)
with the same trained model. Positions with errors are marked by red. The
uninformed corrector struggles with correcting errors in the sampling process.

###  6.2 Gene Expression Data

To test if these sampling schemes also capture correlations on real data, we
use a gene expression time series dataset [36] and the evaluation metrics from
Kerrigan et al. [37]. We quantize the values by binning them into $S=64$
discrete bins and train our model on the discretized data. Kerrigan et al.
[37] evaluates the quality of the generated sequences by computing point-wise
statistics for each position along the sequence lengths and computes the mean
squared error (MSE) of these statistics between the samples and true data.

Unfortunately, these point-wise metrics are ineffective at testing the
correlation structure of individual sampled sequences, since they only measure
marginal distributions on each sequence dimension. However, as we have
demonstrated previously, correctors are needed to compensate for the
predictor’s failures to capture dependencies in the data. In order to better
test this, we calculate the point-wise metrics on the discrete cosine
transform (DCT) of the sequences.

The results are shown in Figure 3. Each method uses 56 model evaluations and 2
corrector steps per predictor step. We find that $k$-Gillespie’s still
reliably outperforms $\tau$-leaping, while the MPF informed corrector yields
the best performance on many metrics.

Figure 3: MSE for different pointwise statistics between the data and model
samples, as compared to the baseline ($\tau$-leaping with the forward-backward
corrector). These statistics are evaluated on the discrete cosine transform of
the sequence to better reflect correlation structure. The mean and standard
deviations are shown for 20 runs of 500 samples each.

###  6.3 Conditional Generation on Monophonic Music

Following Campbell et al. [9], we test conditional generation by modeling
songs from the Lakh pianoroll dataset. In this dataset, monophonic sequences
of length 256 are extracted, with each coordinate representing one of 128
notes or a rest. Campbell et al. [9] trained a discrete denoising diffusion
model with the uniform forward process. The training objective was altered to
target the last 224 time steps of the input sequence given the first 32. We
construct our hollow transformer model with approximately the same number of
parameters and train it for over 1M iterations.

We report our best results in Table 1 . The Hellinger distance is computed
between the histograms of the generated and ground truth notes, and outliers
are counted with the notes that are in the samples but not in the ground
truth. We see that absorbing diffusion combined with $k$-Gillespie’s and the
MPF corrector again generates high quality samples with only a fraction of the
computational cost.

Table 1: Results on the monophonic piano dataset. The metrics are computed over the test set with 5 samples for each test sequence. The baseline results are taken from Campbell et al. [9]. For our results we use $\tau=0.02$ and $k=3$.  Method | NFEs | Hellinger Distance | Proportion of Outliers
---|---|---|---
$\tau$-leaping + 2Uninformed (uniform) | 2800 | $0.3762\pm 0.0015$ | $0.1091\pm 0.0014$
$\tau$-leaping + 1Informed:MPF | 95 | $0.3769\pm 0.0009$ | $0.1231\pm 0.0011$
$k$-Gillespie’s + 1Informed:MPF | 142 | $\mathbf{0.3744\pm 0.0003}$ | $\mathbf{0.1071\pm 0.0010}$

##  7 Discussion

We proposed informed correctors and modified algorithms for discrete diffusion
models with absorbing forward processes. We showed that common scheme of
$\tau$-leaping with the forward-backward corrector is not well-suited to this
setting. To address this problem, we develop novel, informed correctors and
provided a modified training strategy to learn them. We showed that this
approach produces superior sample quality with fewer model evaluations than
previous methods. Overall, our work serves to further enhance the capabilities
of diffusion models in discrete regimes.

Limitations Using the informed correctors for absorbing diffusion requires
architecture modifications such as using the hollow transformer. In practice,
we cannot apply the correctors directly for pretrained models and have to
retrain them from scratch. This limits the direct applicability of our
methods.

Future work In developing the informed corrector, we observed that design
decisions in the training and sampling phases can be strongly coupled. To
leverage better sampling methods, the need for modifying the training setup
naturally arises. Future work could explore this coupling in more depth, for
example, by studying how the architecture and training objective can be
further adapted to efficient sampling strategies.

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##  Appendix A Unifying the TauLDR and score entropy objectives

We first reproduce the TauLDR objective:

$\displaystyle\mathcal{L}(\theta)$
$\displaystyle=\int\mathbb{E}_{q_{t}(y)r_{t}(z\mid
y)}\left[\left\\{\sum_{x^{\prime}\neq
y}\nolimits\tilde{R}_{t}^{\theta}(y,x^{\prime})\right\\}-\left\\{\sum_{z^{\prime}\neq
y}\nolimits
R_{t}(y,z^{\prime})\right\\}\log(\tilde{R}_{t}^{\theta}(z,y))\right]\,dt$ (14)
$\displaystyle=\int\mathbb{E}_{q_{t}(y)r_{t}(z\mid
y)}\left[\tilde{R}_{t}^{\theta}(y)-R_{t}(y)\log(\tilde{R}_{t}^{\theta}(z,y))\right]\,dt,$
(15)

where $R_{t}(y)\triangleq-R_{t}(y,y)=\sum_{z^{\prime}\neq
y}R_{t}(y,z^{\prime})$. Here $r_{t}$ represents the probability of a specific
transition, given that a transition happens at time $t$:

$r_{t}(z\mid y)=\begin{cases}\frac{R_{t}(y,z)}{R_{t}(y)}&y\neq z\\\
0&y=z\end{cases}$ (16)

Campbell et al. [9] rewrote the joint distribution :

$q_{t}(y)r_{t}(z\mid y)=\sum_{x_{0}}q_{0}(x_{0})\psi_{t}(z\mid
x_{0})\phi_{t}(y\mid z,x_{0}),$ (17)

and then analytically integrate out $y$ to reduce sampling variance, given
factorization assumptions on $R_{t}$. The downside of this approach is that
the resulting loss becomes very complicated and unintuitive.

To circumvent this complication, we propose to instead rewrite the objective
with a label switching trick:

$\displaystyle\mathcal{L}(\theta)$
$\displaystyle=\int\mathbb{E}_{q_{0}(x_{0})q_{t}(y\mid x_{0})r_{t}(z\mid
y)}\left[\tilde{R}_{t}^{\theta}(y)-R_{t}(y)\log(\tilde{R}_{t}^{\theta}(z,y))\right]\,dt$
(18) $\displaystyle=\int\mathbb{E}_{q_{0}(x_{0})q_{t}(y\mid
x_{0})}\left[\tilde{R}_{t}^{\theta}(y)-\sum_{z\neq
y}R_{t}(y,z)\log(\tilde{R}_{t}^{\theta}(z,y))\right]\,dt$ (19)
$\displaystyle=\int\mathbb{E}_{q_{0}(x_{0})q_{t}(y\mid
x_{0})}\left[\sum_{z\neq y}\tilde{R}_{t}^{\theta}(y,z)-\sum_{z\neq
y}R_{t}(y,z)\log(\tilde{R}_{t}^{\theta}(z,y))\right]\,dt$ (20)
$\displaystyle=\int\mathbb{E}_{q_{0}(x_{0})}\left[\sum_{y}\sum_{z\neq
y}\left\\{q_{t}(y\mid x_{0})\tilde{R}_{t}^{\theta}(y,z)-q_{t}(y\mid
x_{0})R_{t}(y,z)\log(\tilde{R}_{t}^{\theta}(z,y))\right\\}\right]\,dt$ (21)
$\displaystyle=\int\mathbb{E}_{q_{0}(x_{0})}\left[\sum_{y}\sum_{z\neq
y}\left\\{q_{t}(y\mid x_{0})\tilde{R}_{t}^{\theta}(y,z)-q_{t}(z\mid
x_{0})R_{t}(z,y)\log(\tilde{R}_{t}^{\theta}(y,z))\right\\}\right]\,dt$ (22)
$\displaystyle=\int\mathbb{E}_{q_{0}(x_{0})q_{t}(y\mid
x_{0})}\left[\sum_{z\neq
y}\left\\{\tilde{R}_{t}^{\theta}(y,z)-R_{t}(z,y)\frac{q_{t}(z\mid
x_{0})}{q_{t}(y\mid
x_{0})}\log(\tilde{R}_{t}^{\theta}(y,z))\right\\}\right]\,dt$ (23)
$\displaystyle=\int\mathbb{E}_{q_{0}(x_{0})q_{t}(y\mid
x_{0})}\left[\sum_{x\neq
y}\left\\{\tilde{R}_{t}^{\theta}(y,x)-\tilde{R}_{t\mid
0}(y,x)\log(\tilde{R}_{t}^{\theta}(y,x))\right\\}\right]\,dt,$ (24)

where $\tilde{R}_{t\mid 0}(y,x)\equiv R_{t}(x,y)\frac{q_{t}(x\mid
x_{0})}{q_{t}(y\mid x_{0})}$ is the true reverse transition rate conditioned
on $x_{0}$. Note that we switched the labels $y$ and $z$ on line (22). We also
renamed $z$ to $x$ on the last line to preserve alphabetical order of the
variables for readability. Now we have a simple Monte Carlo estimator for this
objective: we just need to sample $y$ from the marginal distribution, and then
evaluate the parameterized transition model $\tilde{R}^{\theta}_{t}(y,x)$ for
all neighbors $x$ of $y$, which can be done using only one model evaluation.

Note again that we can use the score parameterization for the reverse process:

$\tilde{R}_{t}^{\theta}(z,y)=R_{t}(y,z)s^{\theta}_{t}(z,y).$ (25)

Plugging this parameterization into 24 and absorbing constant terms into $C$,
we get the diffusion weighted denoising score entropy objective (Theorem 3.6
of Lou et al. [13]):

$\mathcal{L}_{DSE}=\int\mathbb{E}_{q_{0}(x_{0})q_{t}(y\mid
x_{0})}\left[R_{t}(x,y)\sum_{x\neq
y}\left\\{s^{\theta}_{t}(y,x)-\frac{q_{t}(x\mid x_{0})}{q_{t}(y\mid
x_{0})}\log(s^{\theta}_{t}(y,x))\right\\}\right]+C\,dt.$ (26)

##  Appendix B Pseudocode for $k$-Gillespie’s Algorithm

We give pseudocode for the $k$-Gillespie’s algorithm presented in Section 3.1
in Algorithm 1. Note that here we use the generation operator $x=G^{d}_{n}(y)$
to represent the sequence obtained by changing the $d$th component of $y$ to
the token $n$.

Algorithm 1 $k$-Gillespie’s Algorithm with Corrector Steps

1:$\hat{R},R^{c},t_{min},\theta$, number of updates $K$, number of corrector
steps $C$

2:Initialize time $t\leftarrow 1$

3:Initialize sample $x\leftarrow\textsc{Mask}^{D}$

4:for $j=1$ to $D$ do

5: Compute backward rate $r^{d}_{i}=\hat{R}^{\theta}_{t}(x,G^{d}_{i}(x))$

6: Calculate total rate $r^{d}=\sum_{i\neq x^{d}}r^{d}_{i}$

7: Sample holding time $\tau^{d}\sim\text{Exp}(r^{d})$

8: // Make multiple state transitions

9: for $k=1$ to $K$ do

10: Get dimension of transition $d^{*}=\textsc{Sorted}(\tau^{d})[k]$

11: Update state $x^{d^{*}}\leftarrow\text{Cat}(\mathbf{r}^{d^{*}})$ where
$\mathbf{r}^{d^{*}}=\frac{1}{r^{d^{*}}}(r^{d^{*}}_{1},\dots,r^{d^{*}}_{S})$

12: end for

13: Update time $t\leftarrow t-\tau^{d^{*}}$

14: break if $t\leq t_{min}$

15: // Apply corrector steps

16: for $c=1$ to $C$ do

17: Compute corrector rate $r^{c}=R^{c}_{t}(x,\cdot)$

18: Apply corrector $x\leftarrow\textsc{PossionUpdate}(x,r^{c})$

19: end for

20:end for

21:Find most likely values for the ungenerated dimensions
$x\leftarrow\arg\max_{x_{0}}p^{\theta}_{t}(x_{0}\mid x)$

22:return $x$