Learning Single Index Models with Diffusion Priors
Authors: Anqi Tang
Youming Chen
Shuchen Xue
Zhaoqiang Liu
Venue: ICML 2025 poster (2025)
Keywords: Single index models
diffusion priors
inversion of diffusion models
Primary area: general_machine_learning->representation_learning

Abstract: Diffusion models (DMs) have demonstrated remarkable ability to generate diverse and high-quality images by efficiently modeling complex data distributions. They have also been explored as powerful generative priors for signal recovery, resulting in a substantial improvement in the quality of reconstructed signals. However, existing research on signal recovery with diffusion models either focuses on specific reconstruction problems or is unable to handle nonlinear measurement models with discontinuous or unknown link functions. In this work, we focus on using DMs to achieve accurate recovery from semi-parametric single index models, which encompass a variety of popular nonlinear models that may have {\em discontinuous} and {\em unknown} link functions. We propose an efficient reconstruction method that only requires one round of unconditional sampling and (partial) inversion of DMs. Theoretical analysis on the effectiveness of the proposed methods has been established under appropriate conditions. We perform numerical experiments on image datasets for different nonlinear measurement models. We observe that compared to competing methods, our approach can yield more accurate reconstructions while utilizing significantly fewer neural function evaluations.

Learning Single Index Models with Diffusion Priors
Anqi Tang" Youming Chen
Shuchen Xue 23
Zhaoqiang Liu'
Abstract
Rauhut, 2013; Scarlett & Cevher, 2021):
y Ax" +
(I)
Diffusion models (DMs) have demonstrated re- markable ability to generate diverse and high- quality images by efficiently modeling complex data distributions. They have also been explored as powerful generative priors for signal recovery, resulting in a substantial improvement in the qual- ity of reconstructed signals. However, existing research on signal recovery with diffusion mod- els either focuses on specific reconstruction prob- lems is unable handle nonlinear measure- ment models with discontinuous or unknown link functions. In this work, we focus on using DMs to achieve accurate recovery from semi -aramerree single index models, which encompass a vari- ety of popular nonlinear models that may have discontinuous and unknown link functions. We propose an efficient reconstruction method that only requires one round of unconditional sam- pling and (partial) inversion of DMs. Theoretical analysis on the effectiveness of the proposed meth- ods has been established under appropriate condi- tions. We perform numerical experiments on im- age datasets for different nonlinear measurement models. We observe that compared to competing methods, our approach can yield more accurate reconstructions while utilizing significantly fewer neural function evaluations.
where A E Rmxx is the measurement matrix, x" E R. is the signal be recovered, e E R is the additive noise vector, and y R is the observed vector.
Although the linear measurement model utilized in com- pressed sensing can serve as good testbed for demonstrat- ing conceptual phenomena, in many real-world problems, it may not justifiable or even feasible. For instance, the binary measurement model employed in 1-bit compressed sensing (Boufounos & Baraniuk, 2008) has attracted sub- stantial interest. This is due to its cost-effective and efficient hardware. It has also been shown that 1-bit compressed sens- ing is robust against certain nonlinear distortions (Laska & Baraniuk, 2012). The limitations of the linear data model have prompted the study of general nonlinear measurement models. Among these, the semi parameereer single index model (SIM) is arguably the most popular (Horowitz, 2009). The SIM models the observations follows:
y f(x)),
(2)
where A consists of i.i.d. standard Gaussian entries, and ʃ R + E is an unknown and possibly random nonlinear link function that is applied element-wise. The aim is to estimate the signal x² using the knowledge of A and despite the unknown nonlinear link f. It is well-known that in the SIM, x" is generally not identifiable (Plan & Vershynin, 2016), since any scaling of x can be incorporated into the unknown f. Consequently, it is common to impose the identifiability constraint || ||2 1.
1. Introduction
To enable efficient and reliable recovery of the underlying signal x", the predominant approach is to introduce struc- tural modeling assumptions, such as sparse and conventional generative priors. For instance, the research works following (Bora et al., 2017) typically achieve signal reconstruction under the assumption that the underlying signal lies within the range of variational autoencoder (VAE) a generative adversarial network (GAN) (Van Veen et al., 2018; 8; Hand et al., 2018; Heckel & Hand, 2019; Wu et al., 2019; Asim et al., 2020; Ongie et al., 2020; Whang et al., 2020; Jalal
The objective of compressed sensing is to accurately re- cover the underlying nigh-dimensional sparse signal from small number of linear measurements, with the measure- ment model as follows (Candes & Wakin, 2008; Foucart &
'Equal contribution University of Electronic Science and Tech- nology of China UUniversity of Chinese Academy of Sciences AAcadmmy Mathematics and Systems Science, CAS. Correspon- dence to: Zhaoqiang Liu <zqliu12e gma.lemmmaa.............
Proceedings of the ndnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
et al., 2021b; Nguyen et al., 2021; Liu et al., 202 lb; 2022a; Genzel et al., 2022; Liu et al., 2024).

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Learning Single Index Models with Diffusion Priors
Compared conventional generative models like VAßs and GANs, diffusion models (DMs) (Sohl-Dickstein et al., 2015; Song & Ermon, 2019; Ho et al., 2020) have recently demon- strated their superiority in terms of robust generative capa- bilities and learning complex data distributions (Dhariwal & Nichol, 2021). DMs have been integrated into a wide variety of domains, leading to transformative progress and reshap- ing traditional approaches in areas such as vision and audio generation (Saharia et al., 2022b; Kong et al., 2021), and reinforcement learning (Chi et al., 2023; Mansen-Esttuch et al., 2023). In particular, when it comes to addressing signal recovery problems, diffusion models offer more flexible and robust solution, facilitating high-quality recon- structions and improved performance (Kawar et al., 2022; Whang et al., 2022: Saharia et al., 2022ca;; Alkan et al., 2023; Chung et al., 2023a; Cardoso et al., 2023; Wang et al., 2023; Chung et al., 2023b; Feng & Bouman, 2023; Rout et al., 2023; Wu et al., 2023; Aali et al., 2024; Chung et al.,
specific linear signal recovery problems or are incapable of handling nonlinear measurement models with discontinuous unknown link functions. Perhaps most relevant to the present paper. the work (Meng & Kabashima, 2022) uses SGMs to learn the underlying distribution of natural signals. By estimating the conditional score function, it enables better reconstruction quality when dealing with quantized compressed sensing problems. Nevertheless, the approach in (Meng & Kabashima, 2022) is restricted to quantized measurements and is relatively slow in reconstruction. Even if utilizing the knowledge of the link function, it requires thousands of neural function evaluations (NFEs) to achieve a reasonably accurate recovery.
1.2. Contributions
The main contributions of this paper are threefold:
We propose an efficient method for accurately recover- ing the underlying signal from nonlinear observations of SIMs by utilizing diffusion priors. Our approach does not require the knowledge of the link function, and it only requires one round of unconditional sampling and (partial) inversion of DMs, resulting in a relatively small number of NFES for the entire process.
2024; Dou & Song, 2024; Song et al., 2024; Wu et al., 2024;
Sun et al., 2024).
Inspired by the strong capabilities of DMs in diverse real- world applications, in this study, we investigate the applica- tion of DMs to learn SIMs.
1.1. Related Work
Based on heuristic theoretical results, we opt to initiate the inversion of DMs from an intermediate time rather than performing the full inversion of DMs. Under suit- able conditions, we provide theoretical analysis on the effectiveness of the corresponding approach. Ad- ditionally. we empirically observe that carrying out partial inversion results in significantly better recon- structions compared to performing full inversion.
Relevant works on signal recovery with conventional gen- erative models (such as GANs and VAEs) have shown that learned priors can drastically reduce sample complexity in compressed sensing and related problems. A comprehensive discussion of diffusion-based and conventional generative approaches is provided in Appendix A. In the following, we highlight diffusion-basse signal recovery methods.
To validate the effectiveness of our method, we con- duct numerical experiments on image datasets under distinct nonlinear measurement models. Notably, in the noisy 1-bit measurement setting, our approach not only achieves substantially faster computational per- formance but also outperforms competing methods in reconstruction accuracy, even when these methods ex- plicitly incorporate knowledge of the link function.
Signal recovery with diffusion models: Diffusion mod- els (DMs) have advanced signal recovery by modeling com- plex data distributions, with two main paradigms: task- specific and pre-trained approaches. Task-specific DMs are trained for particular problems, such as super-resolution (Sa- haria et al., 2022c) or deblurring (Ren et al., 2023). Pre- trained DMs, used without task training, guide re- construction via score estimation (e.g.. (Feng et al., 2023)) or adapt frameworks like DDPM for linear measurements (e.g., DDRM (Kawar et al., 2022)). MCG (Chung et al., 2022a) and HIGMM (Song et al., 2023) are mainly designed for the linear setting. Additionally, IIGMM can be extended to certain nonlinear settings (where the link function is known) by leveraging combination of pseudoinverse op- erations and nonlinear transformations. DPS (Chung et al., 2023b) and DAPS (Zhang et al., 2024a) are applicable in nonlinear settings as well, also primarily under the assump- tion that the link function is known.
1.3. Notation
We use upper and lower case boldface letters to denote matrices and vectors respectively. For any M E N, we use the shorthand notation [M]] {1, 2.......My) MM. and we use IM denote the identity matrix in MxM Given two sequences of real values {a:} and {b,}, we write a, ((b))
if there exists an absolute constant Ci and positive integer i such that for any i > i1, |a, < C16b. We use S to denote the unit sphere in IR", i.e., 5n- = {s E R"
|||||| = 1}. For a generator G : R= + R", we use R(G) to denote the range of i.e. R(G) = (G(=) : 2 E R*).
However, the works mentioned above either concentrate on
2

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Learning Single Index Models with Diffusion Priors
2. Preliminaries for Diffusion Models
By substituting V log (x) 1//o into Eq. (7) and taking advantage of its semi-linear structure, we ob- tain the following numerical integration formula (Lu et al., 2022a; Zhang & Chen, 2022):
Diffusion Models (DMs) define a forward process (0j/7 starting from = 0 and hp. For any & [0, T], the distribution of 2, conditioned on zo satisfies the follow- ing equation:
xt = eli f(r)d Holha x. o./
(eli (el strarr g2(T) 20,
Eo co(x7, dr.
goe (or(2;//0)) = (o of I.).
(3)
(9)
Here, a and are non-negative, differentiable functions possessing bounded derivatives. Moreover, de is monoton- ically non-increasing with ag = 1, while o, is monotoni- cally increasing. It has been demonstrated that the following stochastic differential equation (SDE) has the same transi- tion distribution goe ( (zo) for any t []] (Song et al., 202 bb; Kingma et al., 2021):
Alternatively, we can use data prediction network o t) that satisfies the condition cœ(2 (x; 0, (zt, t))/o, (Kingma et al., 2021; Salimans & Ho, 2022). By substituting Eq. (5) into Eq. (9) and using the
transition between Eg and zg. we obtain the following inte- gration formula (Lu al., 2022b):
Tt 2t
_xx E, + o J/m² eo A)d2, 0x
(10)
dæ; f())x,dt + g(()dw,, 3o (x0))
(4)
where X, == log(as//// /or) is strictly decreasing with respect
where w, E I" is the standard Wiener process, and qu represents the marginal distribution of æ4. Moreover, f(t)
to t and has an inverse function (-).). Thus, xe (z,, t) can be written as ng (A)) = + A).
a scalar function known the drift coefficient and g(t) is scalar function called the diffusion coefficient, which are
2.1. The Sampling Process of Diffusion Models
specified as follows:
The sampling process in DMs gradually denoises a noise vector to an image vector that approximately follows the same distribution as the training data. More precisely, if we divide the time interval [e, T) into N sub-intervals with iN tN-1 < ... < f1 to T, and use first-order numerical scheme for Eq. (10), for i E [N. we can derive the following iterative formula for the transition from time - to tr:
f(t) dlog at
gt) do? d dloga a oi.
(5)
dt
dt dt
Under certain regularity conditions, Song et al. (2021b)
demonstrate that the forward process in Eq. (4) has an equiv- alent reverse process running from time T to O, which starts with the marginal distribution (#T):
dæ, = [f()) - Vz) a log qe (xx)]dt g(t)du, (6)
ot. :t.., + to 01,-1
(at. Oct, Or, 0r,-1
xo xo (eo(
4-11;
where W. E R" represents the Wiener process in reverse time, and V. log ge (2:)) stands for the time-varying score function.
(11)
which is in line with the widely-used DDIM sampling method (Song et al., 2021a).
For more efficient sampling, Song et al. (2021b) further show that the following probability flow ordinary differen- tial equation (ODE) possesses the same marginal distribu- tion at each time Ê as that of reverse SDE in Eq. (6):
Moreover, second- or third-orda numerical schemes can be employed to enhance the sampling efficiency further (Lu et al., 2022b; Zhao et al., 2024). Specifically, the second- order multi-step numerical solver for Eq. (10) yields the following iterative formula for i 2 2 (Lu et al., 2022b):
dæ: = f(t)x, - 2 (t) (x.), Ey ~ 9r (WT). dt
Tt. -t.-. at2 (e-hi -1)
Dt.
(7) The only unknown term in Eq. (7) is the time-varying score function V_ log qe (z.). In practice, DMs typically estimate the scaled score function -o, x log q: (æx) by using noise prediction network e (0 t), and optimizes the parameter through minimizing the following objective:
01,11
x (1
1
xo (i)
1
xe ( t--2)
2r;
2ri
(12)
where h, > - X and r; h4-1/h, We refer to the corresponding sampling method as DM2M for brevity.
Exeg go go (o. [I + oiE,t) E112] dt.
'In practice, it is common to set the end time of sampling as small > 0 instead of 0 avoid numerical instability, as described in (Luu et al., 2022a, Appendix D.

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Learning Single Index Models with Diffusion Priors
Let Hy be the function corresponding to the i-th sampling step. For instance, for the DDIM sampling method in Eq. (11), we have
different from Eq. (16):
. 04 Lt, + 0ts- 04111 0n
xo (oo 44).
On
0:-1 Oai
(17)
ai 0:11 <((x)= == 0n x + ot. 0e,
xe (2, 4-1))
Furthermore, similar to DM2M in Eq. (12), by considering the second-order numerical discretization of Eq. (10) from E to t T, we can obtain the corresponding iterative formula for the inversion of DMs.
Oz, 0ta-1
(13)
Then, the entire sampling process can be expressed as ã. G(āT), where the generator G : R" + R is the composition of of K1, R2........ KN, i.e.
Letting eú be the function corresponding to the i-th inversion step. Then, the entire inversion procedure can be written as Ky = G (Φ.). where the inversion operator Gt : R + R" is the composition of 01, U2,..., UN. i.c.,
(xx) KN . (1))).
(14)
In addition, for convenience, for any & (e, we use G2 to denote the sampling process from to (and thus GT = G). Specifically, if letting in = max E [N] 44-1 2 t), we have that G2 is the composition of Ka,, Ra, +1, KN. i.e.,
G (z) G1 01 02 uN(x))
(18)
In addition, for convenience, for any E [e, T), we use G to denote the inversion from to T (and thus G Gt).
Specifically, if letting jr min(j E [M] 1j (), we have that G is the composition of of U3., uj.., Un ie.
Gz(z)) = KN o...... k o Ki (z).
(15)
2.2. The Inversion of Diffusion Models
c = un 0 0 o (z).
(19)
The inversion in DMs retraces the ODE sampling process with the aim of identifying the specific noise vector that gen- erates a given image vector. This inversion is a crucial aspect in various applications, including image editing (Hertz et al., 2023; Kim et al., 2022; Wallace et al., 2023; Patashnik et al., 2023), watermark detection (Wen et al., 2023), and image- to-image translation (Kawar et al., 2022; Su et al., 2023). Recently, the inversion of DMs has attracted increasing at- tention, and several relevant methods have emerged (Pan al., 2023; Zhang et al., 2024b; Wallace al., 2023; Hong et al., 2024; Wang et al., 2024).
3. Setup and Approaches
Recall that the nonlinear observation vector y E R" is assumed to be generated in accordance with the SIM in Eq. (2), where x=" 8n1 is the signal to be estimated. We follow (Liu & Liu, 2022) to assume that the unknown link function f satisfies the following conditions:
/ ;= Ea-N( [s (a (0Tx))a a² a²=²"] #0,
(20)
E Ea-((..
[ (a (a"x')" <0..
(21)
For example, the naive DDIM inversion method (Hertz et al., 2023; Kim et al., 2022) has the following iterative proce- dure:"
The conditions in Eqs. (20) and (21) are mild and hold true for various forms of f. such as f(x) = sign(x) for 1-bit measurements and f = x3 for cubic measurements. However, as noted in prior works such as (Liu & Scar- lett, 2020a; Liu & Liu, 2022), the condition in Eq. (20) is not satisfied for phase retrieval models with f(x) = x² or f(z) = |x|| (or their noisy versions).
01: 2t, + 0t.-
a1 1 xe (t., 111).
Ctt
Tti
Ot:-1 Tt:
(16) The naive DDIM inversion method circumvents the compu- tational overhead of the implicit numerical scheme by sub- stituting 2e ( with xe (ir,,, 1)). It has been widely used in imaging applications like image editing. Al- ternatively, by taking into account the first-order numeri- cal discretization of Eq. (10) from t = e to t = T. it is straightforward to derive an iterative formula that is slightly
Remark 1. The condition in Eq. (20) is classic and crucial condition for SIMs. For example, it is (albeit implicitly) assumed in the seminal work (Plan & Vershynin, 2016) and in subsequent research that builds upon it. If this condi- tion fails to hold, specifically when μu = O, the recovery of μa= as (Plan & Vershynin, 2016) and in our Eq. (24) below becomes meaningless. Additionally, we follow (Liu & Liu, 2022) to assume the condition in Eq. (21), which generalizes the assumption that s(a x'' )) is sub-Gaussian (which is satisfied by quantized measurement models), and accommodates more general nonlinear measurement mod- els such as cubic measurements with f(x = x²3 and their noisy counterparts.
FFor higher-order sampling methods, we can derive analogous representations for G in Eq. (14) and G. in Eq. (15) using re- cursive formula. An illustration this process can be found in Lemma within Appendix C.
Forr brevity, we assume that the sampling and inversion of DMs utilize the same time steps. In practice, they might employ different time steps and have a varying number of steps.
4

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Learning Single Index Models with Diffusion Priors
We utilize generator G R Ie that corresponds to the sampling process of DMs (cf. Eq. (14)) to model the un-
Lemma 2. Under conditions for the link function f in Eqs. (20) and (21), we have the following holds w.h.p.:
derlying signal. Specifically,, we assume that x² E R where R(G) denotes the range of G. This assumption is
C' "VVlg(22) (2g)
A y px
(24)
standard for nonlinear measurement models with generative priors and has also been made in prior works such as (Wei et al., 2019; Liu & Scarlett, 2020a).
vm
where C' is sufficiently large positive constant.
Let G* : R + IR be an operator corresponding to an inversion method for DMs (ef. Eq. (18)). The work (Liu & Liu, 2022) sets the estimated vector as Par(A" y/m), where Pg denotes the projection operator onto the range of G. Since performing the projection step can be time- consuming even for conventional generative models like GANs, the authors of (Raj et al., 2019) use the composition of the pseudoinverse of the generator and the generator itself to approximate the projection operator. Building on these works, to reconstruct the direction of the signal x" from the knowledge of the measurement matrix A E mR and the observed vector E R (despite the unknown link f), a natural idea is to set the estimated vector & as follows:
The comparison between Lemmas and 2 inspires us to express _ A as ma
1 AA mp
C' Cuym
(25)
for E ~ IO,,.,) Additionally, from Eq. (3), we know that ot can be written as 3t = a; + O,E, = at (2o + (or/aa)e,) with e, ~ ~N.O..,,) Since the nonlinear link function f is unknown (and consequently, μ is also un- known), and C, C' are undetermined positive constants, we employ a tuning parameter C, and find a time !* such that
Or"
C. Vm'
m ATy) (1ATA)
(26)
a Go
(22)
ate
and start the inversion from time 8* with the input vector being Ctg- O/A'A y y/m, where C²2 > 0 is tuning parameter. Remark Unlike prior works such as (Chung et al., 2022b; Fabian et al., 2024; Wu et al., 2024) that identify an inter- mediate time for sampling, we seek an intermediate time mainly for the inversion of DMs. Moreover, this intermedi- ate time is computed using theoretical findings in Lemma 2.
However, such simple idea does not fully exploit the re- lationship between ! m ATy and the underlying signal x", and it leads to unsatisfactory empirical results for diffusion models, as we will see in Section 5. In particular, in the approach corresponding to Eq. (22), the inversion step starts from time e (for an e close to 0; see Section 2.1). and this implicitly assumes that 1 A y approximately follows the m
target data distribution qo- However, based on the following lemmas, observe that 1 AA",, y should instead be regarded m as noisy version of x= ~ go. especially when the number of samples m is relatively small compared to the ambient dimension n. First, it is easy to obtain the following lemma.
Then, the corresponding estimated vector is
â = ( (at=C' m y
)
(27)
Lemma 1. Iffe ((,,L,,, I,), then we have that the following holds with high probubility'
where Gf- denotes the partial inversion operator (from t* to T; see Eq. (19)). One may also notice that since J A Tuy can be regarded as noisy version of x and the sampling process has a denoising effect, we can simply sample from t" to without taking the inversion step. In this case, the corresponding estimated vector is
|| I C. Co/log(2m),
(23)
where C is a sufficiently large positive constant.
Furthermore, we have the following lemma, which demon- strates that 1 AA7 y is a noisy version of μæ=" (see Eq. (20) m for the definition of μ4) and approximately characterizes the
(c. (amC(aAT) m
a =
(28)
noise level. The proofs of Lemmas 1 and 2 are placed in Appendix B.
where G₁- denotes the partial sampling operator (from t* to e; see Eq. (15)). Indeed, such a simple and efficient approach can also lead to reasonably good reconstructions, although its performance is inferior to that of the approach corresponding to Eq. (27).
"This assumption can also be approximately expressed as x²" mu Go when G corresponds to an excellent diffusion modeling of the target data distribution qu- The assumption that x" go is typically implicitly assumed by works that utilize diffusion models to solve compressed sensing problems.
We refer to the approach for Eq. (22) as SIM-DMFIS (SIMs using diffusion models with full inversion and sampling
'Here and in the rest of the paper. a statement is said to hold with high probability (w.h.p.) if it holds with probability at least 0.99.
'If
Co er set t. to T, and Vm or
Co < 26, set t" to Vm o,
5

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Learning Single Index Models with Diffusion Priors
procedures), the approach for Eq. (28) as SIM-DMS since it only performs (partial) sampling procedure, and the approach for Eq. (27) as SIM-DMIS. We illustrate the dif- ferences among these three approaches in Figure 1. For convenience, we present the SIM-DMIS approach in Algo- rithm 1.
Lemma 3 shows that it is reasonable to assume that G is Lipschitz continuous. The proof of Lemma 3 and the precise characterization of the Lipschitz constant L for DDIM and DM2M are deferred to Appendix C.
Then, we have the following theorem regarding the effective- ness of the SIM-DMIS approach as presented in Eq. (27). Specifically, for t [e, T", it offers an upper bound for the distance between G G (ix) and Ct. where H. is a sample of q: and i. is on the same ODE trajectory as Ut. When e sufficiently small and the (scaled) data prediction network De provides good approximation the ground-truth score V log Q: (Chen et al., 2023d; Li et al., 2023), E. will be close to the ground-truth data following zo that corresponds to G. Additionally, it has been demonstrated in (Lu et al., 2022a:b) that under certain regularity conditions, DDIM and DM2M are first- and second-order numerical sampling methods for Eq. (10), respectively. Moreover, by consider- ing the first- and second-order numerical discretization of Eq. (10) from t = to t = T. we similarly obtain the first- and second-order numerical inversion methods for Eq. (10) respectively (for instance, the inversion method in Eq. (17) is first-order numerical inversion method).
(a) SIM-DMS
(b) SIM-DMFIS
(c) SIM-DMIS
Figure 1. An illustration our three approaches. For SIM-DMS, we only perform the sampling from t" to For SIM-DMFIS, we perform the full inversion and sampling procedures. For SIM- DMIS, first perform the inversion from ' to T, and then perform the sampling from I to c.
Algorithm 1 The SIM-DMIS approach
Input: A E Rmxxn y E Rmm Ho. time steps c = ix < IN-1 <... < 4 < to- = T, generator G corresponding to the sampling process as in Eq. (14), partial inversion operator Gi as in Eq. (19), tuning parameters C2 and C²
1: Calculate t* [c. 7] as in Eq. (26).
Therefore, based on Theorem 3, if the term Oi- ve-CC,AA T y/m in Eq. (27) can be approximately expressed as a,, + o with e being a standard normal vector, d == G G (ay-C,,A" T y //mm is close to x" under appropriate condi- tions. The proof of Theorem 3 can be found in Appendix C.
2: Calculate the estimated vector a as in Eq. (27).
Output: a
4. Theoretical Analysis
Theorem 3. For E [e, T), let Dt E R" be sample of qu and let C = T; + o. Sx. ez A)dA, which is the analytic solution of Eq. (10) with respect to the initial vector De. For kI, k2 E 21, suppose that G corresponds to ky-th order numerical inversion method for Eq. (10), and G denotes the partial inversion operator from to T (see Eq. (19)). Suppose that the generator G corresponds to a k2-th order numerical sampling method for Eq. (10) and G is L-Lipschitz continuous. Then, under certain regularity conditions, we have the following:
In this section, we provide a theoretical analysis of the ef- fectiveness of our proposed SIM-DMIS approach. Before presenting the analysis, we first present some preliminary results. Finstly, the following assumption states that the neu- ral function Na (z, 1) is Lipschitz continuous with respect to its first argument. This assumption has been widely adopted in relevant works such as (Lu et al., 2022a; Chen et al., 2023a;c; 2024; 2023d; Bortoli, 2022; Lee et al., 2022; Li et al., 2023).
Assumption 1 (Lipschitz Continuity of the Neural Func- tion). For any t E [0. 21. there exists a positive con- stant L, > O such that the neural function 22(2, t) is Li-Lipschitz continuous with respect to its first param- eter x. That is, holds for all 2,, 272 E Raa that [x)) 20 l)l S L, - x2l12-
|| - GG G = =0 (Vrl (Niin + Lh (i (29)
where hynax = maxte(N] - -t;-r).
5. Experiments
Based on Assumption 1, it can be casily demonstrated that the generator corresponding to popular sampling methods such as DDIM and DM2M (cf. Sec. 2.1) i) is Lipschitz contin- uous. Specifically, we have the following lemma.
In this section, we conduct series of experiments to vali- date the effectiveness of the proposed (see Algorithm 1). Specifically, we evaluate our method on two datasets: FFHQ 256 x26 (Karras et al., 2019), Ima- geNet 256 256 (Deng et al., 2009). 8 For FFHQ and Im-
approach
Lemma 3. Suppose that the data prediction network xo satisfies Assumption 1. Then, if G : IK + RD is the generator as in Eq. (14), we have that G is L-Lipschitz continuous with with L > 0 being dependent on {au,) NN1 fa,, ) and (
"These conditions are similar to those in (I.u et al., 2022a, Appendix B. and are listed in Appendix C. DDue the page limit, the results of CIFAR-10 (Krizhevsky & Hinton, 2009) are included in Appendix D.
6

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Learning Single Index Models with Diffusion Priors
ageNer, the ambient dimension is n 3 256 256 196608.
variance exploding (VE) schedule" for the FFHQ dataset. As for the other methods, employ the model from DPS, which has been trained on 49,000 FFHQ 256x256 images and validated on the first 1,000 images.
We perform experiments on nonlinear measurement mod- els with 1-bit and cubic measurements, for which we have = -ign Axx + e) and y = (Ax*)² ) + e respectively, where e ~ (O, ((0,,"Im) Im). We compare S TIM-DMIS with OCS-SGM proposed (Meng & Kabashima, 2022), which uses DMs to handle quantized compressed sensing prob- lems. We also compare our approach with the posterior sampling methods DPS (Chung et al., 2023b) and DAPS (Zhang et al., 2024a). DPS is a popular baseline for general signal recovery problems, and DAPS has shown excellent performance in nonlinear signal recovery problems. When DPS and DAPS utilize the knowledge of the link function f." the corresponding methods are denoted as DPS-N -N and DAPS-N respectively. In addition, we compare with DPS and DAPS for the linear operator without using the knowl- edge of f, and the corresponding methods are denoted as DPS- L and DAPS- L respectively.
It is evident from Table and Figure that SIM-DMIS ex- hibits superior reconstruction performance with 1-bit mea- surements. Notably, SIM-DMIS requires only 150 NFEs, in contrast to QCS-SGM, which demands a significantly higher 11,555 NFEs. Additionally, both DPS and DAPS utilize 1,000 NFEs. The results for cubic measurements presented in Table 2 and Figure 3 further demonstrate the effectiveness of our proposed methods. In particular, SIM-DMIS $ consis- tently achieves the highest reconstruction quality across all evaluated metrics while requiring only 150 NFEs, demon- strating both the efficiency and robustness of our approach under the cubic measurement setting the FFHQ dataset.
Note that in 1-bit measurements, even in the noiseless sce- nario, the link function f(x) sign(x) is non-differentiable at x = O (though as mentioned in Footnote 9, Py- Torch can still enforce automatic differentiation). The differretiiaaility also poses challenges for DPS-N and DAPS S they rely on f in gradient based updates. This can lead to inaccurate gradients and ultimately resulting in subpar performance." 12 For cubic measurements, since we have observed from the results for 1-bit measurements that STM-DMFTS, DPS-I, and DAPS-L do not perform well, we do not compare against them. Furthermore, we do not include comparisons with DAPS-N since although the link function for cubic measurements is differentiable, its pronounced non-linearity produces unstable gradient direc- tions that make optimization with DAPS- -N unreliable and significantly degrade reconstruction performance.
For the QCS- method, we adhere to the experimen- tal settings specified in (Meng& & Kabashima, 2022) to en- sure a fair comparison. Additionally, we compare with SIM-DMFIS and SIM-DMS (see Eqs. (22) and (28)), which are two variants of SIM-DMIS. We employ three metrics to assess the performance of these methods: Peak signal-to-noise ratio (PSNR), structural similarity index measure (SSIM), and learned perceptual image patch sim- ilarity (LPIPS)) The reported quantitative results are av- eraged over 100 testing images. In the experimental pro- cess, use DDIM for sampling and the DM2M inversion method (refer to Sec. 2.1 for details). The reported NFEs rep- resent the total number of NFEs utilized by each approach. For instance, in the case of S STM-DMTS S and $ SMMDMMF T S, the reported NFE is the sum of the number of steps in the inversion process and the number of steps in the sampling process. We adopt the variance preserving (VP) schedule use 50
Table 1. Quantitative results for FFHQ with 1-bit measurements (m n/8 = 24576). We mark bold the best scores, and underline the second best scores.
with 8iin = 0.1 and 8max 20. For SIM-DMS NFEs, while for SIM-DMIS S and SIM-DMFIS DMFIS, we use 150 NFEs." 10 The remaining parameters are set as = 0.001,
Method
NFE
PSNR (1)
SSIM (t)
LPIPS (L)
T = 1. and a = 0.05. For the detailed parameter settings of C. and C" in SMMDMMS and SIM-DMIS, please refer to Appendices F and G.
DCS-SGH
11555
12.9142.36
0.51
0.50+00.08
DPS-N
1000
11.14#1.46
0.37
0.69 0.05
DPS-L
1000
8.57+2.05
0.22+0.08
0.69
DAPS-N
1000
16.5910.54
0.33+0.05
0.48+0.055
DAPS-I
1000
5.63+
0.04
0.60
Results for FFHQ (256x256) Since the experiments for FFHQ are time-consuming, we only report the results for m n1/8 24576. For the QCS- SGM method, we lever- age pre-trained unconditional Score-SDE model with the
SIM-DMS
50
17.1442.41
0.4440.07
0.48+0.05
SIM-DMFIS IS
150
0.48+
0.03
0.90+0.02
SIM-DMIS
150
19.87+2.77
0.60
0.37
For 1-bit measurements where f is not always differentiable, PyTorch can still enforce automatic differentiation.
"l "nttss//huggingface.c htt.:/ggiiiggggcooooooo//////eee wwwgooole//essnn--ffhleee.o. 255
1h same number of NFEs is used for the inversion process of S.M--MMMM DMMDDIIS-AMM S and s SIM-DMFTS. As demonstrated in Appendices F and G, this choice has negligible impact on the experimental results.
12 supplementary material of DAPS suggests using Mctropo- lis Hasting for non-differentiable forward operators, but it is also mentioned to have inferior performance and low efficiency, with results only in the supplementary material.
7

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Learning Single Index Models with Diffusion Priors
Results for ImageNet (256x250) For ImageNet, we re- port the results for n//16 12288. We utilize a pre- trained conditional ImageNet 256x256 model along with its corresponding classifier sourced from ADM (Dhariwal & Nichol, 2021). We adhere to the default recommended configuration settings of these models, substituting the un- conditional ImageNet 256 x x256 model initially provided by DPS. 13 The tests are conducted on 100 images selected from the Validation K Set. 14 Given that QCS-SGM lacks pre-trained models for ImageNet, we do not compare with it.
Table 2. Quantitative results on the FFHQ with cubic measure-
ments (m = n/8 24576). We mark bold the best scores, and
underline the second best scores,
Method
NFE
PSNR (t) SSIM (t) LPIPS (L)
DPS-N
1000
1..4
0.37+00.8 0.50++0.07
SIM-DMS
50
14.4412.11
0.51+0.10 0.47+0.08
SIM-DMIS
150
17.56:22.56
0.57=00.0 0.40=10.07
Table 3 and Figure 4 demonstrate that SIM-DMIS achieves superior reconstruction with 1-bit measurements compared to DPS and DAPS variants, while requiring only 150 NFEs. The results with cubic measurements, summarized in Ta- ble 4 and Figure 5, further validate the advantages of our methods Notably, SIM-DMTS achieves the best perfor- mance across all metrics while maintaining low compu- tational cost of only 150 NFEs. These results underscore the effectiveness and efficiency of our approach in handling cubic measurement settings on the ImageNet dataset.
Q2 023-5004
Table 3. Quantitative results for ImageNet with 1-bit measurements (m = n/16 = 12288). We mark bold the best scores, and underline the second best scores.
DMP
Method
NFE
PSNR (t)
SSIM (1) LPIPS ())
DPS-N
1000
11.57:22.3
0.24
0.64
200 1M--04
DPS-L
1000
12.67=3.43
0.11=0.16 0.81
DAPS-N
1000
14.62+1.20
0.12#0.01
0.61
DAPS-L
1000
6.20+1.66
0.08
0.64
STM-DMS
50
16.93+3.16
0.32+0.09
0.54
SIM-DMFIS
150
11.89
0.0810.02
0.76
SIM-DMIS
150
18.06+3.16
0.45+0013
0.46
Figure Examples of I-bit reconstructed images for FFHQ.
Table 4. Quantitative results on the ImageNet with cubic measure- ments (m n/ 16 = 12288). We mark bold the best scores, and underline the second best scores.
Method
NFE
PSNR (1)
SSIM (1) LPIPS (L)
EMG-1 M
DPS-N
1000
11.47+2.97
0,22+0.08
0.62+10.06
SIM-DMS
50
14.99+2.82
0.33 0.59-
S1A DNEE
SIM-DMI
150
16.2843.38
0.40 0.5
EM OVES
19 For DAPS, we follow the settings described in the origi- nal paper and employ a pre-trained unconditional model from ADM (Dhariwal & Nichol, 2021).
Figure 3. Examples of cubie reconstructed images for FFHQ.
8

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Learning Single Index Models with Diffusion Priors
6. Conclusion
In this paper, we present approaches for learning SIMs by making use of the sampling and inversion methods for pre-trained unconditional DMs. Theoretical analysis and numerical results provided illustrate the effectiveness of the proposed approaches.
OPP W
Impact Statement
DPE
This paper presents work aimed advancing the field of Machine Learning, particularly the area of signal recov- ery using diffusion models (DMs). The proposed method applies DMs to enhance the recovery of signals from Single Index Models (SIMs), offering more flexible and robust solu- tions for complex signal reconstruction tasks. It could have broad applications in fields such as imaging, signal process- ing, and data reconstruction, with potential to improve the efficiency and accuracy of various real-world tasks. While direct ethical concerns associated with the method- ology, the potential societal impacts of this work include improvements in computational efficiency and the enhance- ment of technologies reliant on image reconstruction and signal recovery.
DAVE N
50h DNER
Figure 4. Examples of 1-bit reconstructed images for ImageNet.
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A. Extended Related Work
In this subsection, we summarize some relevant works, which can roughly be divided into (i) nonlinear signal recovery with conventional generative models, and (ii) signal recovery with diffusion models.
Nonlinear signal recovery with conventional generative models: The seminal work (Bora et al., 2017) studies linear compressed sensing with conventional generative priors (such GANs and VAEs) and demonstrates via numerical experiments on image datasets that using a pre-trained generative prior can significantly reduce the number of required measurements (compared to that of using the sparse prior) for accurate signal recovery. This has led to significant volume of follow-up works in (Dhar et al., 2018; Hand et al., 2018; Shah & Hegde, 2018; Jagatap & Hegde, 2019; Asim et al., 2020; Ongie et al., 2020; Daras et al., 2021; Liu & Scarlett, 2020b; Jalal et al., b;a), which explore various aspects of high-dimensional signal recovery with generative priors. A literature review in this area can be found in (Scarlett et al., 2022).
In particular, under conventional generative priors, there have been several studies on 1-bit compressed sensing (Qiu et al., 2020; Liu al., 2020; 2021a) and phase retrieval (Hand et al., 2018: 8; Hyder al., 2019; Jagatap & Hegde, 2019; Aubin et al., 2020; Shamshad & Ahmed, 2020; Liu et al., 2021b; Killedar & Seelamantula, 2022; Liu et al., 2022b), among others. Moreover, SIMs that account for unknown nonlinearity and conventional generative priors have been explored in several studies, including (Wei et al., 2019; Liu & Scarlett, 2020a; Liu & Liu, 2022: Chen et al., 2023b). More specifically, the works (Liu& & Scarlett, 2020a; Liu & Liu, 2022; Chen et al., 2023b) provide recovery guarantees for the generalized Lasso approach and practical algorithms under the assumption of Gaussian sensing vectors. In (Wei et al., 2019), under the extra assumption that the link function is differentiable, SIMs with link functions of first- and second-orden are studied through Stein's identity, allowing for general non- onnGaassian sensing vectors.
Signal recovery with diffusion models: Due to the ability model complex distributions, DMs have demonstrated remarkable performance in variety of applications. The seminal work on DMs (Sohl-Dickster) al., 2015) introduces a framework for the diffusion process, which has been further developed in multiple forms, such as denoising diffusion probabilistic models (DDPMs) (Ho et al., 2020) and score-based generative models (SGMs) (Song et al., 2021b). These models have been widely applied to signal recovery problems (Daras et al., 2024), which can be broadly classified into two main paradigms.
One paradigm is task mask-specific, in which DMs are trained for particular tasks. For example, the model in (Saharia et al., 2022c) is customized for super-resolution, while the model in (Ren al., 2023) is developed for deblurring. The other paradigm makes use of pre-trained DMs that are not limited to any specific signal recovery problem. Works within this paradigm can be further classified according to how they handle the measurements. Some approaches use DMs as priors to direct the reconstruction process by estimating the score function, as shown in (Feng et al., 2023; Daras et al., 2022). Other approaches, such as DDRM (Kawar et al., 2022), adapt the DDPM framework specifically for linear signal recovery problems. By incorporating singular value decomposition to limit the solution space, DDRM can effectively handle linear measurements. However, its performance degrades when dealing with sparse or limited measurements, rendering it less effective in challenging situations.
To solve general noisy linear signal recovery problems, some recent works (Chung et al., 2022a; 2023b; Song et al., 2023; Fei et al., 2023; Fabian et al., 2023; Zhang et al., 2023) aim to estimate the posterior distribution using the unconditional diffusion model based on Bayes' rule. In particular, MCG (Chung et al., 2022a) approaches data consistency from the perspective of the data manifold. It proposes a manifold-constrainee gradient to ensure that corrections stay on the data manifold. DPS (Chung et al., 2023b) abandons the projection step in the reverse process because it may cause the sampling path to deviate from the data manifold. Instead, it approximates the gradient of the posterior through a specially hand-designed strategy, where the measurements can be regarded as a signal conditioning the sampling process. 1IGDM (Song et al., 2023) further develops this type of strategy into unified expression that encompasses various signal recovery problems by using the Moore-Pennoss pseudoinverse. IIGDM guides the diffusion process by matching the one-step denoising solution and the ground-truth measurements, after transforming both via a generalized pseudoinverse of the measurement model. DAPS (Zhang et al., 2024a) modifies the traditional diffusion process by decoupling consecutive steps in the sampling trajectory. This allows for greater variation between steps, improving the exploration of the solution space and leading to more accurate and stable reconstructions.

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Learning Single Index Models with Diffusion Priors
B. Proofs of Lemmas 1 and 2
Before presenting the proofs, we present the following simple tail bound for a Gaussian random variable.
Lemma 4. (Gaussian tail bound (Wainwright, 2019, Example 2.1)) Suppose that ~ NNaa, o²) is Gaussian random variable with mean a and variance o². Then, for any u > 0. P al 2 u) S 26
B.1. Proof Lemma
For u > and any j E [nl], from Lemma 4, from Lemma 4, we have with probability at least 1 - 22- that |eyl S u. Taking a union bound over all E [n], we obtain with probability least 2ne that |Eeloo S u. Then, by setting C (lg((2n), we obtain the desired result.
B.2. Proof Lemma
Let gi = a x" Since E Sn1 we know that gi are independent standard normal random variables. Additionally, note that
1 pa
1 m
A y
y:a,
pax
(30)
m
m
1
Σ
V:(a; (a x )=²)
Σ
Vi (a x: )xx
uax"
(31)
1=1
1=1
For any j [n]. since [[a.99:] = 23. aij can be written as
0 = zjg + V (xz)²
(32)
where rij is a standard normal random variable that is independent of ga. Then, the j-th entry of the first term in the right-hand side of Eq. (31) can be written
y (a: (a, x
(aTxx")x") )x²)
1
Σ Va Vi 9:27))
(33)
m
V (zg)²
yelly-
(34)
m
Note that yi [(af x²)= f()). Thus riy is also independent of yi- Let M2 == (o and & be the event that 1 Dm 1==12 v < 2M2. From (Liu & Liu, 2022, Lemma 3), we have
p(E < C1
(35)
where C is positive constant depending on Moreover, conditioned the event E,, the term Em VaTay in Eq. (34) zero-mean Gaussian with the variance being Em1 v². Then, from Lemma 4, we obtain
is
P
yo'
< 2exp
2 2
2exp
4m.M2a
(36)
Taking union bound over all E [n]. we obtain with probability at least 1 - 2n 2x (
n²
that
imdi,
1 m
m = = =|| 1 Σ
Σ
(a, (a x= )a=))
(1z)2
yrg
(37)
m
m
m
Furthermore, the j-th entry the second term in the right-hand side Eq. (31) can be written as
m2
1
m
We Va xx" )x="
1ex"
Ua vi x0
11 x.
(38)
7
m7
2

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Learning Single Index Models with Diffusion Priors
For any 7 > O, let Eg be the event that 1 m Σi vi x²) p < From (Liu & Liu, 2022, Lemma 3), we have
P(E)) v C mT
(39)
where Cjj is positive constant depending on f. Additionally, conditioned on &2, we have
(
- < Σ
m
yi (a x²) - x =
vi a x²) a ||000| 11x0 ST.
(40)
Combining Eqs. (31), (35), (37), (39), and (40), observe that by setting u Ci (1vlog(2n) v and
C2 /ogloo(2 for some sufficiently large positive constants C1 and C2 and letting C" C1 + C2. we obtain the desired result.
C. Proof of Theorem? 3
First, we provide a proof for the auxiliary result in Lemma 3 for popular sampling methods such as DDIM and DM2M (cf.
Sec. 2.1). Specifically, we have the following detailed version of Lemma 3.
Lemma 5. Suppose that the data prediction network 2e satisfies Assumption Then, if R - R" is the generator corresponding to the entire sampling process of DDIM (see Eq. (11)), we have that G I.-Lipschitz continuous with
L L=
II i=1
Tt + oe, 01,-1
Ctt ot.
1 L...). On-1
(41)
Additionally, suppose that G RR" Rd the generator corresponding to the entire sampling process of DM2M (see Eg. (12)), and suppose that for = 0, 2,, ..., Tt, Lı-Lipschitz continous with respect uto- Then, we have that Ta can be set and L1 can be set to be + o aio recursive formula:
asa
Leg and for L; can be calculated via the following
aio
L, ( -
Tle [h 1a-+a((1-0 h (1+
2r: _ Lum. LI-1 + aa, e1-e 1 2r;
-1-
(42)
In particular, we have that G is L-Lipschitz continuous with L = LN.
Proof. For DDIM, by the triangle inequality, we have the following inequality for any i E [N] and any æ 002 E IR":
(1s(2))-<((<))ll) - (2,2))l ( Ot. + dt. -
( (tt. 0z, ot.-1 ) .L...) || 1|111110]) x2l122
ar,-1
(43)
where Kę is defined in Eq. (13). Additionally, it is easy to show that if function f is L. -Lisschit and is Ly -Lpishhchi then their composition f is (T, -Lischhizz (Bora et al., 2017). Then, we obtain that the generator G = nN
Ki
(see Eq. (14)) is L-Lipschitz continuous, with L given by Eq. (41). Note that for the first sampling step of DM2M, we need to use the first-order numerical scheme- DDIM. -DenM Then, similarly to the case of DDIM, based on Eq. (12), we can obtain the desired recursive formula for the Lipschitz constants.
Then, we list the regularity conditions on æg required for Theorem as follows. These conditions are similar to those in (Lu al., 2022a, Appendix B..
The total derivatives
A dx7
(as a function of X) exist and are continuous for j = 0,
The neural function æ, t) is Lipschitz continuous with w.r.t. its first parameter æ (i.e., Assumption 1).
hmax
(aX >o._.) (11NN).
Based on the above auxiliary results, we are now ready to present the proof of Theorem 3.
3

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Learning Single Index Models with Diffusion Priors
C.1. Proof of Theorem 3
Let Gf be the analytic inversion operator (corresponding Eq. (10)) from time to T. That is, for any R", we have
[// AT e A)dA.
(x) oT x + or 0t
(44)
Additionally, let & be the analytic sampling operator (corresponding to Eq. (10)) from time T to That is, for any x E IR",
we have
de S/ c' A)dA.
0. G(z)=
(45)
oT
Let äT = G] = a. tt + OT SN e l0xx A)dd. We have
Go G( = G(ãT)
(46)
% - or + de //m E (oo AldA
(47)
dr
NN
de
OT Is tor frv po
co 2)dA + toe //m (tto(ds)
(48)
OT
f'r dr
e' 1)dA + f/m de
Oe Di
de
(49)
Ot
de
>x
de
Oe 1i -
e² c2200(xx)))ll
(50)
d1
he
(51)
Therefore, we obtain
- a. (CC
(52)
< [[C Go (z))
(53)
< [ - (((ra))))(((()))a((((((()))(<))) +
(54)
where the last inequality follows from the L-Lipschitz continuity of G and कn (wx). Under the regularity conditions listed above, similarly to (Lu et al., 2022a, Theorem 3.2), we have
(x.) (2:))12 On
(55)
and
([(#T)-C(TTIIl G(T))ll2 OVThh
(56)
Combining Eqs. (54), (55), and (56), we obtain the desired result.
D. Experimental Results for CIFAR-10 with 1-bit and cubic Measurements
In this section, present the experimental results obtained from the CIFAR-10 dataset (Krizhevsky & Hinton, 2009) using 1-bit and cubic measurements. For the CIFAR-10 dataset, the ambient dimension is 32 x 32 x 3 3072, and the pixel values are normalized to the range [0, 1]. Prior experimental findings have shown that the SIM-DMIS method outperforms STM-DMS and STM-DMF TS in terms of reconstruction performance. Therefore, in this study, we limit our comparison to cannot handle
our SIM-DMIIM approach against OCS-SGBD DPS- N, and DPS-L- For 1-bit measurements, since cubic measurements, we compare against OneShot proposed in (Liu & Liu, 2022), which uses GANs solve nonlinear signal recovery problems. Additionally, since we have observed from the results for 1-bit measurements that SIM-DME IS does not perform well, we do not compare against it. The experimental results demonstrate the superior performance of our proposed method SIM- DMIS across different datasets and measurement settings.
Note that the term O in the statement of (Lu al., 2022a, Theorem 3.2) is added element-wise. Thus when considering the 6 norm, we have an extra ni factor, where n the data dimension.

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Learning Single Index Models with Diffusion Priors
Results C- (32x32) with 1-Bit Measurements For 1-bit measurements, report results under two surement settings: m 500 and 1000, corresponding to approximately 16% and 32% of the ambient dimension We utilize pre-trained unconditional DDPM model with the variance preserving (VP) schedule for CIFAR- 10. Prior experimental findings have shown that the S SIM-DMIS method outperforms S SIM-DMS and SIM-DMF IS in terms of reconstruction performance. Therefore, in this study, we limit our comparison to our $ SIM-DMIS S approach against QCS- SGMM, DPS- N, and DPS-L, L. As shown in Table 5, SIM-DMIS consistently outperforms existing approaches across both m = 500 and m 1000. Qualitative results in Figure 6 further confirm strong reconstruction capability.
Table 5. Quantitative results for 1-bit with 500 and 1000 measurements. We mark bold the best scores, and underline the second best scores.
Method
NFB
m
500
m = 1000
PSNR (1)
SSIM (1)
LPIPS (4)
PSNR (1)
SSIM (1)
LPIPS (4)
QCS-SCM
1160
8.40+:3.75
0.25:+0.1 0.55+0.20
11.76:4..9 0.41
0.41
DPS-N
1000
9.43
0.04
9.70+2.19 0.57+:0.55
0.04
0.65
DPS-L
1000
11.53
0.08 0.61
12.2442.71
0.10 0.6910.07
SIM-DMIS
150
16.24+1.65
0.55++.100 0.40+0.10
18.61
0.65+
0.31
Origin
Dest
MMM-DMM
SINA
Figure 6. Examples of reconstructed images for CIFAR-10 with 1000 1-bit measurements.
Results on CIFAR-10 (32x32) with Cubic Measurements The quantitative results are shown in Table 6, and examples of reconstructed images presented in Figure 7, consistently outperforms OneShot across all measurement
5

---

Learning Single Index Models with Diffusion Priors
settings. Specifically, with 500 measurements, SIM-DMIS achieves approximately 15% higher PSNR and 2% better SSIM compared to OneShot. With 1000 measurements, SIM-DMIS achieves approximately 25% higher PSNR and 12% better SSIM compared to OneShot. Furthermore, SIM DMIS S consistently yields lower LPIPS value, indicating better perceptual similarity to the original images, compared to OneShot
Table Quantitative results for CIFAR-10 with cubic measurements. We mark bold the best scores, and underline the second best scores.
500
1000
Method
PSNR (1)
SSIM (1)
LPIPS (J)
PSNR (t)
SSIM (t)
LPIPS (L)
OneShot
13.68:12.099
0.33110.09
0.50100.4
14.61::2.09
0.3910009
0.48-10.04
SIM-DMIS
15.80.12.02
0.45110.10
0.43
16.9712.21
0.5511.12
0.38
Origin
Figure 7. Examples reconstructed images for CIFAR-10 with m 1000 cubic measurements.
E. Additional Examples of Reconstructed Images for FFHQ and ImageNet with 1-bit Measurements
In this section, we present some additional examples of reconstructed images for the FFHQ 256x256 dataset in Figure and also present some additional examples of reconstructed images for the ImageNet 256x256 dataset in Figure 9.
6

---

Learning Single Index Models with Diffusion Priors
SGANDER
ORAMIEL
OBAMIEA
Figure Examples of reconstructed images for FFHQ m n/8 24576 1-bit measurements, Top: Origin, Middle: SIM-DMS, Bottom: SIM-DMIS.

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Learning Single Index Models with Diffusion Priors
Origin
DMS
SIM-
SIM
TDMIS SIM
Figure 9. Examples of reconstructed images for ImageNet with n/16 12288 1-bit measurements.

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Learning Single Index Models with Diffusion Priors
F. Ablation Study for SIM-DMS
In this section, we present a detailed analysis of the ablation experiments conducted on SIM-DMS using the CIFAR-10 10 dataset with m 500 1-bit measurements. The experimental setup follows the same framework as in the main body of the research, and the sampling technique is applied throughout.
The results shown in Table 7 provide valuable insights into the performance of SIM-DMS under different parameter configurations. Specifically, we analyze the impact of hyperparameters Cs. C²+, and the number of function evaluations on model performance.
When varying C, while fixing C² = 55 and NFE = 50, we observe a clear upward trend in both PSNR and SSIM, indicating improved signal quality and better structural similarity with increasing C.. However, I.PIPS reaches its minimum value at moderate C., suggesting that the best perceptual quality is achieved with balanced parameter settings. On the other hand, when C, is fixed at 55 and C², is varied, we find that lower values of C²2 yield better performance across all metrics, with PSNR and SSIM reaching their peak at the lowest tested value. LPIPS remains relatively stable but shows slight decrease at lower C, values. Finally, when we adjust the NFE with C. 55 and C² = 55, all performance metries show minimal fluctuation, suggesting that once NFE reaches a certain threshold, increasing it further does not yield significant improvements.
In conclusion, these results underscore the flexibility of SIM-DMS with respect to parameter tuning. The parameter C, demonstrates clear impact on reconstruction quality, while lower values of C² tend to yield better results. Additionally, the stability across different NFE values indicates the model's efficiency even with fewer iterations. These findings provide valuable guidance for parameter selection in practical applications of S IM-DMS.
Table Ablation experiments for SIM-DMS on CIFAR-10 with 500 1-bit measurements, with different values of C., C² and NFEs.
We mark bold the best scores, and underline the second best scores.
C, (C, 55)
C² (C, 55)
NFE (C, 55, C.
55)
Value
50
52
55
58
60
50
52
55
58
60
20
50
80
100
PSNR (1)
14.441
14.625
14,860
14.956
14.971
15.337
15.214
14,860
14,451
14.205
14,860
14.860
14.860
14.860
11.033
11.19
11.20
112275
11.33
1.354
112255
1:220
11.185
11.166
11.001
11.20
110201
1:1.00
SSIM (1)
0.427
0.442
0.460
0.467
0.470
0.481
0.475
0.460
0.440
0.426
0.460
0.460
0.460
0.460
1:0096
1:00099
1:0.103
100
10.108
107
1: 104
1:0.103
100101
1:00099
20,103
1:0.103
1:0103
100103
LPIPS (L)
0.444
0.436
0.433
0.441
0.447
0.440
0.436
0.433
0,440
0.447
0.433
0.433
0.433
0.433
+0.000
+0.00
+:0099
200097
20009
100000
20.098
1:0099
::0001
+0,000
200094
#00094
+000
10,00
G. Ablation Study for SIM-DMIS
In this section, we present a detailed analysis of the ablation experiments conducted on SIM-DMIS using the CIFAR-10 10 dataset with a measurement of m = 500. The experimental setup follows the same framework as in the main body of the research, and the DDIM sampling and DM2M inversion technique is applied throughout. The results are shown in Table 8.
When varying C. while fixing C² = 55 and NFE = 50, the trend of PSNR and SSIM are consistent with the SIM-DMS where higher values of C. also led to better signal quality and structural similarity. On the other hand, when C, is fixed at 55 and C², is varied, the trends for PSNR and SSIM are largely similar to the s SIMDDMS results, where lower values of C² lead to better performance in these metrics. Interestingly, while the improvement in PSNR and SSIM is clear, LPIPS remains relatively stable across a wide range of C², values, with a slight decrease at lower values of C" particularly around 50 and 52. Finally, when adjusting the NFE with C, = 55 and C² 55, the results show minimal fluctuation in all performance metrics. This observation is consistent with the findings from SIM-DMS, where similar behavior was noted.
In summary, the findings from the ablation study of SIM-DMIS underline several important insights. The parameter C, has a notable impact on both the signal quality and structural similarity of the reconstructed images, with higher values leading to better PSNR and SSIM. The parameter C², plays a crucial role in enhancing PSNR and SSIM, with lower values yielding better results, while LPIPS remains stable. Additionally, the minimal effect of increasing NFE suggests that SIM-DMIS is highly efficient and provides stable performance even with fewer evaluations.
9

---

Learning Single Index Models with Diffusion Priors
Table Ablation experiments for $ S on CIFAR-10 with m 500 1-hit mcasurements. We mark bold the best scores, and
underline the second best scores.
C, (C²- 55, NFE=150)
C² (C, 55, NFE=150)
NFE (C, 55, C. 55)
Value
50
52
55
58
60
50
52
55
58
60
100
150
200
250
PSNR (1)
15.388
15.562
15.733
13.809
15.820
16.247
16.091
15.733
15.322
15.057
15.733
15.733
15.733
15.733
11.211
11:31
1:1420
1..92
1.6655
11.500
1:1355
1.3307
11.420
1:1420
SSIM (1)
0.489
0,506
0.522
0.531
0.534
0.548
0.540
0.522
0,500
0.484
0.522
0.522
0.522
0.522
1:0097
100098
1:0.099
1:0.00
10.104
110,101
1:0099
1:0.099
1:0.099
1:0,099
210,09
110.099
1:0.99
100099
LPIPS (L)
0.406
0.399
0.402
0,411
0.417
0.409
0.405
0,402
0,404
0,411
0.402
0.402
0.402
0,402
20,05
20000
+0.005
20.103
20.008
20.05
2:0003
2:000
20.095
20095
10005
H. Comparative Evaluation of SIM-DMS and SIM-DMIS on Performance and Efficiency
In this section, we evaluate the performance and efficiency of SIM-DMS and SIM-DMIS on the FFHQ dataset. The experimental setup follows the same framework as the main body of the research. The results presented in the main paper (Table 1) show that SIM-DMIS consistently outperforms QCS- SGM and 9IM-DMFIS in both reconstruction quality and NFE. Notably, QCS-SGM requires over 1,0000 NFEs to achieve competitive results, incurring high computational cost. Due to the performance of IM DMIS s over both methods, and the particularly high computational cost of OCS-SGM, exclude SCM and SIM-DMFI IS from the comparisons in this section.
Performance Evaluation As discussed in Footnote 10 and Appendices FF and G, ablation studies on CIFAR-10 show that further increasing the NFE for SIM-DMS does not lead to substantial performance enhancements.
To further verify this behavior and to compare the relative performance of SIM-DMS and IIMDMMIS we conduct additional experiments on the FFHQ dataset. The results, summarized in Table 9, indicate that under the same NFEs, SIM-DMIS consistently outperforms S IM DMS across all standard metrics.
Table Quantitative results on FFHQ (m n/8 24576). mark bold the best scores, and underline the second best scores.
Method
NFE
PSNR (t)
SSIM (1)
LPIPS (4)
FID (4)
DPS-N
1000
11.14=1.46
0.37+00009
0.69+0.05
349.24
DPS-L
1000
8.57+2.05
0.22+-0.88
0.69+00.99
109.01
DAPS-N
1000
16.59:0.54
0.3310005
0.48
138.70
DAPS-L
1000
5.63=0.71
0.04+0003
0.61
322.75
SIM-DMS
50
17.14=2.41
0.44
0.48=0.05
105.04
SIM-DMS
150
17.72+2.63
0.46+0008
0.48
95.52
SIM-DMIS
50
18.78+3.09
0.5810.10
0.41
89.47
SIM-DMIS
150
19.87=2.77
0.60+10.09
0.37
76.21
Reconstruction Speed We conduct additional experiments to measure the reconstruction speed of SIM-DMS and SIM-DMIS. The inference time is averaged over 10 validation images from the FFHQ dataset. All experiments are exccuted on a single NVIDIA GeForce RTX 4090 GPU. As shown in Table 10, the results indicate that SIM-DMS and SIM-DMIS exhibit significantly faster reconstructions when compared to the competing methods.
10

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Learning Single Index Models with Diffusion Priors
Table 10. Comparison of inference time (in seconds) for reconstructing 10 images FFHQ We mark bold the best scores, and underline the second best scores.
Method
NFE
Inference Time (s) (L)
DPS-N
1000
142
DPS-L
1000
142
DAPS-N
1000
160
DAPS-L
1000
160
STM-DMS
50
1.96
SIM-DMIS
150
5.66


Reviews:
[{"review_id": "BWOOoRz6te", "reviewer": "Reviewer_H7Q4", "summary": "Summary\nThis paper proposes a novel method for reconstructing images from measurements obtained through a nonlinear compressed sensing model. The degradation model consists of a measurement matrix, an unknown and potentially discontinuous nonlinear element-wise link function, and additive Gaussian noise. \n\nThe proposed reconstruction algorithm begins by initializing the estimate using the measurement vector, applying the transposed measurement matrix, and applying an empirically tuned normalization. The final reconstruction is achieved using a pre-trained diffusion model (DM). The method involves partial DM inversion followed by DM sampling, where the inversion start time is determined based on the norm of the DM inversion initialization. The authors provide a theoretical analysis by proving a theorem that offers an upper bound on the distance between the reconstructed image and the desired image under specific conditions.\n\nMain findings:\n- The authors establish an upper bound for the distance between the reconstructed and the desired images.\n- Experimental results demonstrate that the proposed approach outperforms existing methods in 1-bit and cubic measurement scenarios.\n- The authors show that the combination of partial inversion and sampling yields better reconstruction quality compared to sampling alone or full inversion followed by sampling.\n\n## Update After Rebuttal\n\nI thank the authors for their detailed responses. I respectfully disagree with the author's assumption that the constant $C$ is fixed and independent of the number of inversion steps (i.e., the number of times the denoiser is applied). This assumption leads to an unrealistic conclusion that increasing the number of inversion steps indefinitely would drive the upper bound on the reconstruction error to zero, thereby implying perfect reconstruction. If this were the case, the authors should demonstrate in the experimental section that the reconstruction error consistently decreases toward zero as the number of inversion steps increases. \n\nA more reasonable and realistic assumption is that the denoiser has a fixed Lipschitz constant **per activation**, in which case the overall constant $C$ in the bound would grow with the number of inversion steps.\n\nFurthermore, in contexts involving bounded quantities, such as image pixels constrained within the range $[0, 1]$, expressing an upper bound using capital-O notation such as $O(1)$ is inappropriate. Instead, the upper bound should be stated explicitly as $distance \\le C$, where $C$ is a meaningful, domain-aware constant (in the case of image pixels less than 1). As noted in my original review, proving that the **per-pixel** distance between the reconstructed and desired images is $O(1)$, which allows for an arbitrary constant, potentially even exceeding 1, does not yield a meaningful or non-trivial insight in this context. \n\nGiven the concerns outlined above, I maintain my original recommendation to reject the manuscript.", "questions": "", "limitations": "", "rating": 1, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the feedback. We are pleased that the claims in our submission have been recognized as \"supported by evidence\" and our experiments have been recognized as \"sound\". Regarding your major concern about the practical relevance of Theorem 2, our responses are as follows:\n-  For practical relevance, the parameter $h_{\\max}=\\max_{i \\in [N]} \\big(\\lambda_{t_i}-\\lambda_{t_{i-1}}\\big)$ in Eq. (29) plays an important role. When the number of sampling/inversion steps is large, for instance, on the order of $10^2$ or $10^3$, $h_{\\max}$ is approximately $10^{-2}$ or $10^{-3}$ (if using typical uniform $\\lambda$ steps). Therefore, when the number of sampling/inversion steps is sufficiently large, in the upper bound $C h \\sqrt{n}$ illustrated by the reviewer, the value of $h$ will be much smaller than $1$ (e.g., $h = 0.001$; note that $C$ is a positive constant of order $\\Theta(1)$). This makes the upper bound practically meaningful when image pixels are bounded within the range $[0,1]$.\n- The proof of Theorem 2 (see Appendix B.1) is built upon the proof for Theorem 3.2 in the popular work for DPM-Solver (Lu et al., 2022a) (note that the upper bound in (Lu et al., 2022a, Theorem 3.2) is for each individual pixel, and thus their bound does not have the $\\sqrt{n}$ factor), which is in turn based on classic analyses of local truncation error for numerical ODE solvers, such as those presented in Section 5.6 of (Burden & Faires, 2005). And our Theorem 2 bears similar practical relevance to the theoretical findings presented in these works.\n\n\nRichard L. Burden and J. Douglas Faires, Numerical Analysis, 8th edition, Thomson/Brooks/Cole, 2005.", "claims_and_evidence": "I the claims made in the submission are supported by evidence.", "methods_and_evaluation_criteria": "In my opinion, yes.", "theoretical_claims": "I have a concern regarding the theoretical part of the manuscript. The upper bound presented in the main theorem appears trivial and lacks practical significance. The theorem states that: \n\n$\\|\\|x - x_r\\|\\|_2 = O(\\sqrt{n}(h^1 + Lh^2))$,\n\nwhere $x$ is the desired image, $x_r$ is the reconstructed image, $n$ is the image dimension, $L$ is a Lipschitz constant, and $h^1$ and $h^2$ are diffusion model parameters.\n\nThis result essentially implies:\n\n$\\|\\|x - x_r\\|\\|_2 \\le Ch\\sqrt{n}$,\n\nwhere $C$ is a positive constant and $h =  h^1 + Lh^2$.\n\nHowever, the image pixels are typically bounded within the range [0, 1]. Therefore, the maximum possible Euclidean distance between any two images is already bounded by $\\sqrt{n}$, representing the distance between a completely black and a completely white image. Consequently, the upper bound derived in the theorem does not provide any meaningful or non-trivial insight. \n\nIf the pixel were unbounded, the theorem might offer valuable insights. However, under the current setting, the theoretical contribution seems to lack practical relevance.", "experimental_designs_or_analyses": "In my opinion, the experiments presented in the manuscript are sound", "relation_to_broader_scientific_literature": "The key contributions of the manuscript are related to the literature about image reconstruction using diffusion models.", "essential_references_not_discussed": "I did not come across related works that are essential for understanding the key contributions of the paper but are not currently cited.", "other_strengths_and_weaknesses": "Since the theoretical analysis is a significant component of the manuscript's contribution, I recommend rejecting the manuscript.", "other_comments_or_suggestions": "I have no further comments.", "questions_for_authors": "I have no further questions for the authors.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "8fwTLx6k6E", "reviewer": "Reviewer_68Nk", "summary": "In this manuscript, the authors address a notable shortcoming in current signal recovery techniques based on diffusion models: most existing methods either concentrate on narrowly defined reconstruction tasks or fail to handle nonlinear measurement models with discontinuous or unknown link functions. To tackle this issue, the authors propose an efficient reconstruction strategy that requires only a single round of unconditional sampling and partial inversion of the diffusion models. Their theoretical analysis and experimental evaluations collectively verify the effectiveness of this approach.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your recognition of this paper and the helpful comments and suggestions. Our responses to the main concerns are as follows. \n\n(**FID is widely recognized as a key evaluation metric in diffusion models. Why wasn’t it employed here?**)\nWe carry out additional experiments to report the FID. The results are presented in Table B1 below. Given the time constraint during the rebuttal period, and following the work for DAPS (Zhang et al., 2024), we compute the FID using a set of 100 validation images. The FID results for QCS-SGM, DPS, and DAPS are not included at this stage as their calculation is relatively time-consuming. However, in the revised version, we will incorporate the FID results for all methods.\n\n(**To ensure a fair and comprehensive comparison, I suggest conducting additional experiments, specifically comparing SIM-DMIS with 50 NFE and SIM-DMS with 150 NFE.**)\nThanks for the insightful comment. We have conducted simple ablation studies for SIM-DMS and SIM-DMIS on the CIFAR-10 dataset (mentioned in Footnote 10, and detailed in Appendices F and G), and the results show that further increasing the NFE for SIM-DMS does not lead to substantial performance enhancements. We also perform the suggested experiments on the FFHQ dataset, and summarize the additional results in the following table. The results also indicate that under the same NFEs, SIM-DMIS consistently outperforms SIM-DMS across all metrics.\n\n| Method   | NFE | PSNR          | SSIM          | LPIPS         | FID    |\n|----------|-----|---------------|---------------|---------------|--------|\n| SIM-DMS  | 50  | 17.14 ± 2.41  | 0.44 ± 0.07   | 0.48 ± 0.05   | 105.04 |\n| SIM-DMS  | 150 | 17.72 ± 2.63  | 0.46 ± 0.08   | 0.48 ± 0.06   | 95.52  |\n| SIM-DMIS | 50  | 18.78 ± 3.09  | 0.58 ± 0.10   | 0.41 ± 0.08   | 89.47  |\n| SIM-DMIS | 150 | 19.87 ± 2.77  | 0.60 ± 0.09   | 0.37 ± 0.05   | 76.21  |\n\nTable B1: Quantitative comparison of SIM-DMS and SIM-DMIS on the FFHQ dataset.\n\n(**It would be beneficial to include additional comparisons highlighting the reconstruction speed of the proposed method relative to existing approaches.**)\nWe conduct additional experiments to measure the reconstruction speed of our method, and the results are presented in the table below. The reported inference time refers to the average reconstruction time for 10 validation images from the FFHQ dataset. All of these experiments are executed on a single NVIDIA GeForce RTX 4090 GPU. The results indicate that SIM-DMS and SIM-DMIS exhibit significantly faster reconstructions when compared to the competing methods (we do not compare with QCS-SGM since it is very time-consuming). We will include the comparisons with respect to reconstruction speed in the revised version.\n\n| Method   | NFE  | Inference Time (s) |\n|----------|------|--------------------|\n| DPS-N    | 1000 | 142                |\n| DPS-L    | 1000 | 142                |\n| DAPS-N   | 1000 | 160                |\n| DAPS-L   | 1000 | 160                |\n| SIM-DMS  | 50   | 1.96               |\n| SIM-DMIS | 150  | 5.66               |\n\nTable B2: Comparisons for reconstruction speed on the FFHQ dataset. \n\n(**Some terms in the references need to maintain consistency with the original papers, particularly regarding capitalization.**) \nIn our revised version, we will carefully check all the references and correct all the inconsistent cases to ensure accurate citation formatting.", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "FID is widely recognized as a key evaluation metric in diffusion models. Why wasn’t it employed here?", "theoretical_claims": "I have reviewed the \"Setup and Approaches\" section and did not identify any issues.", "experimental_designs_or_analyses": "I have carefully reviewed the experimental results and noticed that the SIM-DMS method achieves the second-best performance with only 50 NFE, whereas SIM-DMIS requires 150 NFE. To ensure a fair and comprehensive comparison, I suggest conducting additional experiments, specifically comparing SIM-DMIS with 50 NFE and SIM-DMS with 150 NFE. This would help better understand the relative performance under equivalent NFE.", "relation_to_broader_scientific_literature": "This work extends the approach of QCS-SGM by addressing its limitations. While QCS-SGM either focuses on specific reconstruction problems or cannot effectively handle nonlinear measurement models with discontinuous or unknown link functions, the proposed method overcomes these challenges. Moreover, the proposed method achieves more accurate reconstructions with significantly fewer neural function evaluations (NFEs) compared to QCS-SGM.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "Strengths:\n\n1. The paper is clearly structured and well-organized.\n2. The experimental evaluations presented are comprehensive.\n3. The theoretical derivations provided are rigorous and complete.\n\nWeaknesses:\n\n1. While the SIM-DMS method achieves competitive performance using only 50 neural function evaluations (NFEs), the SIM-DMIS approach requires 150 NFEs. To ensure a fair comparison, the authors are encouraged to perform additional experiments by evaluating SIM-DMIS with 50 NFEs and SIM-DMS with 150 NFEs. This would provide a clearer understanding of the relative performance of both methods under an equivalent NFEs.", "other_comments_or_suggestions": "1. It would be beneficial to include additional comparisons highlighting the reconstruction speed of the proposed method relative to existing approaches.\n2. Some terms in the references need to maintain consistency with the original papers, particularly regarding capitalization. For instance, on line 500, \"ImageNet\" should be capitalized exactly as in the original source. Please carefully check and correct all similar cases to ensure accurate citation formatting.", "questions_for_authors": "FID is widely recognized as a key evaluation metric in diffusion models. Why wasn’t it employed here?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "29eqdqTNq9", "reviewer": "Reviewer_g5pJ", "summary": "The authors propose a diffusion model sampling scheme to reconstruct an unknown signal from measurements, assuming a single-index model with a known compressed sensing matrix and noise distribution but unknown and potentially nondifferentiable link function. The approach leverages a property of the link function to find an intermediate time at which to begin inversion, thereby making inversion much more efficient than the naïve approach. The authors provide a theoretical analysis and numerical experiments on image datasets, comparing to state-of-the-art diffusion inverse solvers.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your helpful comments and suggestions. Our responses to the main concerns are as follows. All citations refer to the reference list in the main document.\n\n(**It is strange that DPS and DAPS with knowledge of the link function (i.e., DPS-N and DAPS-N) perform worse than they do without knowledge of the link function (i.e., DPS-L and DAPS-L).**)\nIn 1-bit measurements, even in the noiseless scenario, the link function $f(x) = \\mathrm{sign}(x)$ is non-differentiable at $x = 0$ (though as mentioned in Footnote 9, PyTorch can still enforce automatic differentiation). The differentiability of $f$ matters significantly. As mentioned in Section 1.1, under the strong differentiability assumption on the link function, (Wei et al., 2019) can handle general non-Gaussian sensing vectors. However, most SIMs research assumes Gaussian sensing vectors, like the seminal work (Plan & Vershynin, 2016) and many follow-up studies such as (Liu & Liu, 2022). Without differentiability, extending these to non-Gaussian cases is difficult.\n\nThe non-differentiability also poses challenges for DPS-N and DAPS-N as they rely on $f$ in gradient based updates. This can lead to inaccurate gradients and ultimately resulting in subpar performance. (The DAPS supplementary material suggests using Metropolis Hasting for non-differentiable forward operators, but it is also mentioned to have inferior performance and low efficiency, with results only in the supplementary material.) In contrast, DPS-L and DAPS-L do not utilize $f$, thus allowing for relatively better performance (note that as demonstrated in the work (Plan & Vershynin, 2016), SIMs can be transformed into linear measurement models with unconventional noises).\n\n(**There is little intuitive explanation of Theorem 2 (I could not find a Theorem 2, by the way). And there is no discussion about whether this would provide samples from a Bayesian posterior.**)\nTheorem 2 essentially says that for any sample drawn from the marginal distribution $q_t$, when we perform the inversion process from time $t$ to $T$, followed by the sampling process from time $T$ to $\\epsilon$, the resulting vector will be reconstructed to closely approximate the corresponding ground-truth data from $q_0$ (and lie on the same ODE trajectory as the original sample). In the revised version, we will incorporate this intuitive explanation in the paragraph preceding Theorem 2.\nWe are unsure what the reviewer means by “I could not find a Theorem 2”. We guess that the reviewer might be referring to the proof of Theorem 2, which is available in Appendix B.1.\n\nThe approaches presented in our work are not directly related to sampling from a Bayesian posterior. While it would be interesting to discuss whether our approaches could yield samples from a Bayesian posterior, we believe that such an exploration is beyond the scope of the current work.\n\n(**The authors should include more intuitive explanations. For example, what exactly are the properties in Eqs. 20 and 21 saying? What is the intuition behind Eq. 25 and why it’s different from Eq. 22? Also, the background on diffusion models is too dense and contains a lot of unnecessary information.**) \nThe condition in Eq. (20) is a classic and crucial condition for SIMs. For example, it is (albeit implicitly) assumed in the seminal work (Plan & Vershynin, 2016) and in subsequent research that builds upon it. If this condition fails to hold, specifically when $\\mu  = 0$, the recovery of $\\mu \\mathbf{x}^*$ as in (Plan & Vershynin, 2016) and in our Eq. (24) becomes meaningless. We follow (Liu & Liu, 2022) to assume the condition in Eq. (21), which generalizes the assumption that $f(\\mathbf{a}^T\\mathbf{x}^*)$ is sub-Gaussian (which is satisfied by quantized measurement models), and accommodates more general nonlinear measurement models, such as cubic measurements with $f(x)=x^3$ and their noisy counterparts.\n\nThe intuition underlying Eq. (25) is that (the scaled version of) $\\frac{\\mathbf{A}^T\\mathbf{y}}{m}$ is approximately the ground-truth signal $\\mathbf{x}^*$ (drawn from the target data distribution $q_0$) with an added zero-mean Gaussian noise component. Then, considering the forward process of diffusion models, which gradually adds Gaussian noise to the ground-truth data, performing a full inversion (from time $\\epsilon$ to $T$) and then a full sampling from time $T$ to $\\epsilon$ as in Equation (22) is inappropriate. As mentioned in the paragraph below Eq. (22), this approach implicitly (and wrongly) assumes that $\\frac{\\mathbf{A}^T\\mathbf{y}}{m}$ approximately follows the target data distribution $q_0$ (without taking additive zero-mean Gaussian noise into account). \n\nIn the revised version, we will incorporate the above intuitive explanations. Additionally, we will streamline the background on diffusion models, focusing solely on the equations and concepts that are fundamental to understanding the core ideas presented in this paper.", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "I was unable to check proof correctness.", "experimental_designs_or_analyses": "Yes, for the 1-bit compressed sensing experiments on FFHQ and ImageNet. It is strange that DPS and DAPS with knowledge of the link function (i.e., DPS-N and DAPS-N) perform worse than they do without knowledge of the link function (i.e., DPS-L and DPS-L). This is true for all the results in Tables 1 and 2 except DPS in Table 2.", "relation_to_broader_scientific_literature": "One perspective is that this work extends the work of Meng and Kabashima 2022 to handle unknown link functions. Another perspective is that it adds to the vast literature of diffusion inverse solvers (including DPS, DAPS, and Pi-GDM) with a simple method for a particular type of inverse problem, namely one where the measurements are an unknown/nondifferentiable transformation of compressed sensing measurements.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "[Strengths] The proposed idea is quite simple and makes for an efficient algorithm. It leverages a property of the link function (Eq. 20), so the simple algorithm has some theoretical inspiration behind it.\n\n[Weaknesses] The method makes several shortcuts without strong theoretical justifications for them. For example, the leap from Eq. 27 to Eq. 28 is rather tenuous if it relies on the assumption that the reverse diffusion process will effectively remove the noise from the naïve estimate. Overall, it is difficult to tell how theoretically justified the method is. There is little intuitive explanation of Theorem 2 (I could not find a Theorem 2, by the way), making it difficult to judge its significance and usefulness. And there is no discussion about whether this would provide samples from a Bayesian posterior.\n\nThe presentation of the paper is rather poor. The authors should include more intuitive explanations. For example, what exactly are the properties in Eqs. 20 and 21 saying? What is the intuition behind Eq. 25 and why it’s different from Eq. 22? Also, the background on diffusion models is too dense and contains a lot of unnecessary information. I recommend sticking the equations and ideas that are essential just for the understanding of this paper.\n\nIt is strange that DPS-N and DAPS-N perform so poorly in the numerical experiments. I am curious why the authors think they would perform so poorly compared to SIM-DMS and SIM-DMIS.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "N/A", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "tTvQFvLfn2", "reviewer": "Reviewer_3BCR", "summary": "The work addresses the problem of signal reconstruction in semi-parametric single-index models, where the link function is unknown. They propose a new method relying on parametrizing the signal prior by a diffusion model. Building on the observation that the measurements may be related to noisy versions of the signal, the related noise variance (and corresponding diffusion time) is learned in the method, and the generator of the diffusion process and its inverse used to yield the final estimator. Theoretical learning guarantees are derived. Numerical experiments on the FFHQ and ImageNet datasets are provided, revealing competitivity of the method compared to existing schemes.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your recognition of this paper and the helpful comments and suggestions. Our responses to the main concerns are as follows. All citations refer to the reference list in the main document.\n\n(**To strengthen the paper, it would be beneficial to include a version of Theorem 2 for the SIM-DMS algorithm (28), if feasible. For now, only the SIM-DMIS algorithm is endowed with the theoretical guarantees. Besides, Theorem 2 does not bear entirely on the SIM-DMIS estimator. Would it be feasible to combine (24) and (29) to reach an end-to-end guarantee?**)\nWe are grateful to the reviewer for the valuable suggestions. In the revised paper, we will include a version of Theorem 2 for SIM-DMS. Eq. (29) within the statement of Theorem 2 essentially says that for any sample drawn from the marginal distribution $q_t$, when we perform the inversion process from time $t$ to $T$, followed by the sampling process from time $T$ to $\\epsilon$, the resulting vector will be reconstructed to closely approximate the corresponding ground-truth data from $q_0$ (and lie on the same ODE trajectory as the original sample). Additionally, Eq. (24) (along with Eq. (25)) essentially indicates that  (the scaled version of) $\\frac{\\mathbf{A}^T\\mathbf{y}}{m}$ approximately follows the marginal distribution $q_{t^*}$ for some $t^*$. If we make the (relatively strong) assumption that (the scaled version of) $\\frac{\\mathbf{A}^T\\mathbf{y}}{m}$ precisely follows $q_{t^*}$ for some $t^*$, then combining Eqs. (24) and (29) provides an end-to-end guarantee. We leave the end-to-end guarantee in the case where Eq. (25) only approximately holds to future study.\n\n(**As a minor comment, the related work section could be further clarified, notably the last paragraph of page 2. Currently, it is unclear which work address linear or non-linear settings, and in which works the link function is (un)known.**)\nAmong the methods described in the last paragraph of page 2, MCG (Chung et al., 2022b) and $\\Pi$GDM (Song et al., 2023) are mainly designed for the linear setting. Additionally, $\\Pi$GDM can be extended to certain nonlinear settings (where the link function is known) by leveraging a combination of pseudoinverse operations and nonlinear transformations. DPS (Chung et al., 2023b) and DAPS (Zhang et al., 2024) are applicable in nonlinear settings as well, also primarily under the assumption that the link function is known. We will clarify these in the revised version.", "claims_and_evidence": "The method is benchmarked against concurrent schemes, in terms of performance metrics and compute. Reconstructed images by the evaluated methods are further illustrated in Figs. 2,3.", "methods_and_evaluation_criteria": "To the best of my awareness, the benchmark datasets considered were also selected in previous works on the topic to evaluate the methods. For instance, FFHQ was used in (Meng and Kabashima, 2023), to benchmark the QCS-SGM method. Therefore, they seem relevant for the evaluation, and ensure comparability with some of the previous works.", "theoretical_claims": "I did not check carefully the proofs of the theoretical claims.\n\nTo strengthen the paper, it would be beneficial to include a version of Theorem 2 for the SIM-DMS algorithm (28), if feasible. For now, only the SIM-DMIS algorithm is endowed with the theoretical guarantees. Besides, Theorem 2 does not bear entirely on the SIM-DMIS estimator. Would it be feasible to combine (24) and (29) to reach an end-to-end guarantee?", "experimental_designs_or_analyses": "I did not check the details of the experimental designs.", "relation_to_broader_scientific_literature": "Although I have limited familiarity with the literature, it is my understanding that compared to key previous works such as (Meng and Kabashima, 2023) or (Zhang et al., 2024), the main novelty of the method is its ability to accommodate unknown non-linear link functions. It also displays competitive performance with previously proposed methods. However, it is possible there exists other methods which I may be missing.\n\nAs a minor comment, the related work section could be further clarified, notably the last paragraph of page 2. Currently, it is unclear which work address linear or non-linear settings, and in which works the link function is (un)known.", "essential_references_not_discussed": "I did not identify any essential missing reference.", "other_strengths_and_weaknesses": "The paper is well written, and intuition is provided in support of the method. I have a few clarifying questions, which I included in the Theoretical Claims section. I am overall in favor of acceptance.", "other_comments_or_suggestions": "I do not have further comments or suggestions.", "questions_for_authors": "Some clarifying questions are detailed in previous sections.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The paper proposes a compressed sensing inversion algorithm based on diffusion modeling. Three reviewers thought the paper was good and should be accepted, while one argued for rejection. The basis for rejection of this reviewer was a perceived triviality of a theorem in the paper, but the authors argue convincingly that this is a misunderstanding on the part of the reviewer and is in line with similar results in compressed sensing theory. Considering this, I am inclined to consider the three positive reviews as reflecting the contributions of this paper, and these reviewers appreciated the comparison with state of art diffusion solvers that indicated the proposed inversion algorithm performs well compared to these. 

One suggestion to consider for the paper was to make it a bit more reader friendly by introducing intuitive examples, etc. While I don't consider this to be required for acceptance, it is useful to note that the paper doesn't truly "begin" until well into the third page with an overly extensive review of related work, which in general is unnecessary. As a result, the experiments are crammed into the last page. Considering the extensive experiments (plus more theory) shown in the appendix, I think the authors have done a more comprehensive evaluation than shown in the main paper and they may consider incorporating some of this into the experiments section and reducing the introductory material in order to balance the main paper.

Overall, I'm inclined to recommend acceptance of the paper considering the positive reviews and more extensive experiments included in the appendix. The theoretical results are also in line with what would be appreciated by the ICML audience.
Decision: Accept (poster)