Flow Along the $K$-Amplitude for Generative Modeling
Authors: weitao Du
Jiasheng Tang
Shuning Chang
Yu Rong
Fan Wang
Shengchao Liu
Venue: ICLR 2026 Poster (2026)
Keywords: generative models
frequency transformation
image generation
ai for science
molecule assembly
Primary area: generative models

Abstract: In this work, we propose K-Flow, a novel generative learning paradigm that flows along the $K$-amplitude domain, where $K$ is a scaling parameter that organizes projected coefficients (frequency bands), and amplitude refers to the norm of such coefficients. We instantiate K-Flow with three concrete $K$-amplitude transformations: Fourier transformation, Wavelet transformation, and PCA. By incorporating the $K$-amplitude transformations, K-Flow enables flow matching across the scaling parameter as time. We discuss six properties of K-Flow, covering its theoretical foundations, energy and temporal dynamics, and practical applications. Specifically, from the perspective of practical usage, K-Flow allows for steerable generation by controlling the information at different scales. To demonstrate the effectiveness of K-Flow, we conduct experiments on both unconditional and conditional image generation tasks, showing that K-Flow achieves competitive performance. Furthermore, we perform three ablation studies to illustrate how K-Flow leverages the scaling parameter for controlled image generation. Additional results, including scientific applications, are also provided.

Published as a conference paper at ICLR 2026
FLOW ALONG THE K-AMPLITUDE FOR GENERATIVE MODELING
Weitao Du
Jiasheng Tang'
Shuning Chang'
Yu Rong'
Fan Wang'
Shengehao Liu
Allibaba DAMO Academy
Universityy of California, Berkeley
The Chinese University of Hong Kong
duweitac. dwt @al ibaba inc. com, scl wuuuchkkkhk.. edu. edu.h.. hk
Fourier
Wavelet
PCA
Low-scaing
High-scaling
Figure 1: Generation using K-Flow with three K-amplitude decompositions: Fourier, Wavelet, and PCA.
ABSTRACT
In this work, we propose K-Flow, novel generative learning paradigm that flows along the K-amplitude domain, where & a scaling parameter that organizes pro- jected coefficients (frequency bands), and amplitude refers the coefficients. We instantiate K-Flow with three concrete K K-amplitude transforma- tions: Fourier transformation. Wavelet transformation, and PCA. By incorporat- ing the K K-amplitude transformations, K-Flow enables flow matching across the scaling parameter as time. We discuss six properties of K-Flow, covering its the- oretical foundations, energy and temporal dynamics, and practical applications. Specifically, from the perspective practical usage, K-Flow allows for steerable generation by controlling the information at different scales. To demonstrate the effectiveness of K-Flow, we conduct experiments on both unconditional and con- ditional image generation tasks, showing that K-Flow achieves competitive perfor- mance. Furthermore, we perform three ablation studies to illustrate how K-Flow leverages the scaling parameter for controlled image generation. Additional re- sults, including scientific applications, are also provided.
of such
INTRODUCTION
Generative Artificial Intelligence (GenAl) represents pinnacle achievement in the recent wave of A1. This field has evolved from foundational methods such as autoregressive models (AR) (Radford
2018), energy-based models (Hinton] 2002 Carreira-Perpinat & Hinton 2005 LeCun et al. 2006
Gutmann & Hyvärinen 2010 Song Kingma 2021), variational auto-encoders (Kingmal 2013)
and generative adversarial networks (Goodfellow al., 2014), diffusion model (Ho et al. 2020
Du et al. 2023), to the most cutting-edge flow-matching (FM) framework (Lipman et al. 2022
Liu
al. 2022b Albergo & Vanden-Fsijnden 2022

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Among these, flow matching (FM) stands out as a density transport method that converts initial simple distribution into a complex target distribution through continuous- flow dynamics. For instance, in the context of image generation, FM learns to map random Gaussian distribution to the pixel-space distribution of images. This process, termed continuous flow, is governed by a local- ized k-dependent vector field (or velocity field) and produces fime-dependent density path, which represents the evolution of the probability distribution over time. As a versatile framework, FM can incorporate a diffusion density path, linking it to established methods such as denoising score matching (DSM) (Vincent 2011 Song & Ermon 2019) and the denoising diffusion probabilistic model (DDPM) (Ho al., 2020).
Motivation. Natural data exhibits an inherent frequency structure, with most of its energy con- centrated in the low-frequency bands (Abry et al.] 1995| Van der Schaaf & van Hateren 1996). Reflecting this property, empirical evidence (Dieleman (224)) shows that DDPMs tend to denoise from low to high frequencies when transforming white noise (with a uniform frequency spectrum)
into meaningful data, allowing earlier recovery of low-frequency components (Biroli et al. 2024 Conventional FMs, however, exhibit different path characteristics (Sun et al. 2025), and their fre- quency progression is not quantitatively established (Figure S2). In parallel, from the reconstruction perspective, recent research (Kouzelis et al.] 2025 Skorokhodov et al. (025) advocates for explicit frequency consistency constraints expectation that introducing frequency-awar par with or exceeding that of conventional FMs. These observations point clear opportunity: developing generative models that offer fine-grained control in the frequency domain can open new frontiers both generation quality and applicability, e.g., frequency editing and restoration.
regularization strategy for auto-encoders. This supports the path into generation can achieve generation quality on
Key Concepts. To formalize our approach, we first establish a unified framework that integrates key frequency-relan reeated concepts from the literature, including Fourier frequency analysis and multi-scale transformations. Central to this framework is the introduction of K K-amplitud space, parameterized by scaling parameter k. The scaling parameter 1 is defined as systematic measure for orga- nizing frequency bands (or coefficients) across different physical systems and processes (Cardy) 1996 Luijten & Blöte, 1996. Behan et al. 2017 Bighin et al. [ Within this framework, we define amplitude the norm of coefficients obtained by projecting data onto bases corresponding to different scaling parameters k. forming what we term the K-amplitude space, or equivalently, -attiinn-e space.
Our Method. Such an understanding of scaling parameter and K- K-amplitude space inspires a new paradigm for generative modeling, which we term K Flow Matching (K-Flow). In essence, K- Flow performs flow along the K K-amplitude. There are two main components in K-Flow, and the first is the K-amplitude decomposition. The K-amplitude decomposition encompasses family of transformations through a linear basis in the K --ampliudde space, and in this work, we explore three types: Wavelet, Fourier, and principal component analysis (PCA) decomposition, as illustrated in Figure T Specifically, K-Flow first applies K K-amplitude transformation to project data from the spatial domain into the K K-amplitude space, where we formulate a novel stochastic interpolant that naturally accommodates the hierarchical structure. In Section A we provide comprehensive analysis of K-Flow through six properties, from theoretical foundations (a & b), energy and temporal dynamics (c & d) to practical applications (e & f), with a detailed pipeline illustrated in Figure 2
We demonstrate the effectiveness of K-Flow through extensive experiments on gener- ation tasks. Qualitatively, our ablation studies reveal the model's scaling controllability that aligns with our theoretical motivation, enabling two key capabilities: (1) efficient class-conditional genera- tion with minimal guidance, where class information is only required during low-k inference stages; and (2) unsupervised frequency editing through various discretizations of the scaling parameter k. Quantitatively, K-Flow achieves sfate-of-the-art or comparable performance in both unconditional generation and training-free image restoration tasks across natural image and scientific datasets.
Our Results.
We distinguish "scaling parameters" in the context of parameterization from "scale" in general discussions
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2 BACKGROUND
In this section, we first formalize the concept of K a-amplitude Decomposition, a process governed by scaling parameter k. In the method section, then detail how our generative framework is designed to operate by flowing along this decomposition, progressively reconstructing the signal.
2.1 SCALING PARAMETER k, AMPLITUDE, AND K-AMPLITUDE DECOMPOSITION
Our data generation framework leverages the implicit hierarchical structure of the data manifold. By 'implicit,, we refer to the hierarchical characteristics that emerge when a generalized K -aamplitude decomposition applied, transitioning the representation from the original data space to the К- amplitude space. Illustrations are in Figure 2 More formally, we represent data as signal & Fd - IR,", finite discretization of Rd and I", where this signal function is equivalent vector. For example, In the case of image data, each pixel of one RGB channel can be viewed as a signal mapping from the spatial grid R² to pixel intensity value IR.. Combining the three channels, they form a vector-valued signal from R² to RR.. An alternative approach is to consider data as a high-dimensional vector Rdxim However, treating data as signal functions better fits the decomposition framework in this work.
Without loss of generality, we take dimension position involves the decomposition of a function using a complete basis set {ez))=1> where n can be infinite. We introduce a scaling parameter k; which partitions the set {ei] into subsets: {e: = (et),, each with ng basis. Hence, signal is expressed as:
1 for illustration. A K-amplitude decom-
--Σo ΦR.
(I)
where dhe := E (v) ]ex)eek for Ejk E {ex). Inspired by the concept of frequency amplitude, we refer to the norm of Φe as the K-am K-amplitude. The parameter l is termed the scaling parameter as expect the natural scaling law exists in well-structured data: the amplitude decays as the value of increases (Field 1987). We define K K-amplitude decomposition (or equivalently, K- amplitude transform) J as the map that sends & to the collection Φk. and denote the collection of all (E (Ø Cyk ee,k) as (k). Then,
F(O) = U=)))(())
We further assume that this transform has an inverse, denoted by F-1,
Splitting Probability. Denote the probability of data Pdatas then the transformations F and F-1 induce a probability measure on the associated K K-amplitude space. In particular, we denote the induced splitting probability of dx as p(k) for each scaling parameter k.
In this work, we explore three types of K K-amplitude decomposition: Wavelet, Fourier, and princi- will provide classic example using the Fourier
pal component analysis (PCA). In Section frequency decomposition on the three- three-dimensional space. This example serves to illustrate the con- struction of the scaling parameter ie and K K-amplitude.
2.2 EXAMPLE: FOURIER AMPLITUDE DECOMPOSITION
Suppose the data & : R R, is drawn from a certain function distribution Pdat- The challenge of directly fitting the distribution Paata is often complex and computationally demanding. Fourier frequency decomposition, however, offers powerful technique to address this challenge by trans- forming Φ into the Fourier space or Fourier domain. In what follows, we will use the terms 'space' and 'domain' interchangeably.
By applying Fourier frequency decomposition, express
of its frequency components.
This transformation can potentially unveil the hidden structure within the distribution Pdatas which is not apparent the spatial or time domain, and it is thus beneficial for understanding the underlying patterns in the data manifold. To illustrate, the continuous Fourier transform F data Φ0x y, 2) R33 RR expressed as:
Flo) Ky, Kz)
1
( y. e 2mi(k, 2m((k.rtkyuttee da dz.
33)
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After this transformation, the spatial variables (z, y, 2) converted into frequency variables (Res ky, kz), thereby representing the data in the frequency domain.
Note that the Fourier frequency is characterized by the high-dimensional vector representation (ki,, kiy, kz). For our purposes, we aim to distill the notion of frequency into a one-dimensional scaling parameter. Namely, define the scaling parameter k as the diameter of the expanding
ball in Fourier space: k L k2 + k2. This definition of k provides simple index that cap-
tures the overall scaling parameter of the frequency components in all directions. Moreover, we can decompose the Fourier transform F()} into groups indexed by the scaling parameter k:
F(+(() = U Flo)(ke, ky, ks).
(4)
Intuitively, (k) represents the set of all frequency components that share the same scaling parameter This grouping allows us to examine the contributions various spatial frequencies of when viewed through the lens of frequency k. Furthermore, O₁ is just the summation of F())(K)
On the other hand, we can recover & from F(o), because the Fourier transform is an invertible operation: == F[ø]. Such an invertibility establishes the Fourier transform valid example of K-amplitude decomposition. For discrete data, which inherently possess one highest resolution, the variables (kz, ky. k;)) are situated discrete lattice rather than spanning the entire continuous space. Consequently, the scaling parameter k derived from these discrete components, is itself discrete and bounded.
2.3 FLOW MATCHING
In this work, we primarily focus on the flow matching (FM) generative models and their fami- lies (Lipman et al. 2022; Liu et al., 2022bb Albergo & Handen-Eijndad (022). In FM, the flow
V, is defined by solutions of an ordinary differential equation (ODE) system with time-dependent
vector field
d v.(x) = ve (x)),,
(5)
di
and we focus on the probability transport aspects of V, In particular, the flow provides a means of interpolating between probability densities within the sample space. Suppose V follows an initial probability Po. then for >>> O, V,, induces probability measure pr: (B( = (o(V)(()), where B is a measurable set. Assume that v,, is differentiable, and define a surrogate velocity at time t (2,, 8) using deep neural network with parameter 8. Then the vector field matching loss is
defined as:
fRM = - don
di dx.( - (x..) di
(6)
By aligning the learned vector field with the true gradient field of the frequency decomposition, this loss function ensures robust approximation and reconstruction of the data. Additionally, every inter- polation π (xo, .1) with a continonn interpolating function fe(zo. 21) between probabilities Po and p1 induces a vector field v, through the continuity equation:
(pe(
V. (p((t)))x(<=))
(7)
and i2 is explicitly expressed as: 0 1 Ex(20.2)) @f((r.=1)) Although explicit matching of o via the continuity equation is intractable, flow matching permits a conditional version:
LORM = Ea(so.a1) L
0 Of((zu. 21) u,(2,, o + constant.
(8)
As detailed in Section 3 our framework reinterprets the time variable as the scaling parameter kt Our goal is to construct K aaplitude-respectee T(xo, 21) with differentiable functions fa-
3 METHODOLOGY: K-FLOW
In this section, we introduce K-Flow. It is constructed from the collection of (k) indexed by specific scaling parameter k. As we will demonstrate in Section 3.1 our approach is independent of the specific construction of the invertible transformation F and the explicit definition of k. This flexibility enables to extend to various K-amplitude decompositions.
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K-Amplitude Space
Lon Sculin
Mideo Scaling
High Boul
Dats
Tarsfor
Noke
Kohrpince
Flww Alono
K-Aespttuce Thansfom
Velocity Function
K-Flow Matching
19,-99
Boaling stepe
Figure 2: Pipeline K-Flow. We have ligure input three inverted images after three transfor- mations at different granularities.
3.1 K-AMPLITUDE INTERPOLANTS
According the concept of stochastic interpolants (Albergo et al. 2023 all flow models can be viewed as constructing stochastic paths that interpolate between a known tractable prior distribution and unknown target distribution, including flow matching (Lipman al., 2022). rectified flow (Liu et al. 2022b). and denoising diffusion (Ho et al. 2020). By incorporating the scaling parameter for K K-amplitude decompositions, ually emerges each amplitude component from white noise. Given that & traverses monotonically from zero to maximum value kmax, this process draws natural analogy to continuous normalizing flows. Since we require F to be invertible, we can reconstruct the data once the complete spectrum the K K-amplitude space generated.
can formulate stochastic interpolant that grad-
To build a continuous flow Ve out of Equation [.. we explore two paradigms designing the interpolants: (1) We generalize the original discrete-valued to continuous values; (2) We ensure that the generation flow, which maps the white noise to the real data, remains invertible such that information lost throughout the process. Still taking the three-dimensional signal Φ(I, y, =) and the Fourier transform F(6) as an example, we realize the second ingredient by introducing noise padding for each & and define the discrete flow & as follows:
- (LVRETRFFREA Flo)(ke, k.) (1- (1-LVErerrqE)
S = FF
(9)
where Il is the indicator function that selects K K-amplitude components up the scaling step k. This formulation ensures that for each step k the reconstruction incorporates the relevant K-Flow components of data and pads the rest with noise Here, the noise is independently drawn from a known distribution (e.g., uniform or Gaussian) for each coordinate (kzz ky, k2). Through this construction, da where kmax represents the maximum scaling parameter of data. This limit condition guarantees that as k approaches its maximum value, the reconstructed (p converges to the original data ø. This behavior is pivotal for the accuracy and fidelity of the generative process. Conversely, Φn simply follows the law of tractable distribution.
stochastic interpolant for the data Φ, ensuring that: lim FPk Φ.
Inter-scaling Interpolant. Since most of the data we aim to generate is discrete in nature, the (kr, ky,, k2) values in the K-amplitude decomposition are inherently defined on a lattice. Conse- quently, the derived scaling parameter k: also takes discrete values. This discreteness implies that SPk originally defined only for discrete values of k. However, this discrete flow imposes a lim- itation: we cannot leverage the powerful flow-matching objective the optimization framework, which requires taking derivatives with respect to continuous scaling step
To handle this issue, straightforward approach is to extend PP to continuous k by intra-scaling interpolation. That is, we want continuous flow Vt, where k E [0, KJ and satisfy Vr чa for integer values of k. Let == [k] represent the continuous scaling step, where [k] denotes the

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integer part of k. Then, the differentiable interpolation of Vx is:
VE = 4 LNi+e
[ F(o +1 ( e
I V ( ks) (l(t) -()))))) -)),
(10)
where ja(t) is a bump function such that ud)) = (d1) = and P (²( -1 The antisymmet- ric property of yl (t) ensures that VE is differentiable from ls for all R allowing the flow matching loss and other gradient-based optimization techniques. In Equation (T). we have three components:
1. +K applies to the amplitude components up to the integer part of k.
2. / applies noise padding to components beyond the next integer.
3. /
performs linear interpolation of the intermediate amplitude com- ponents based on the current
Localized Vector Fields. Instead of directly modeling M.. we pivot our focus to its conditional gradient field, d&,, dk By concentrating on the gradient field, we facilitate a dynamic view how
ΦE evolves with respect to k. To derive an analytical expression of dWg dk conditioned on given instance pair of data and noise: (ó, e), in what follows, we assume that F is a linear transform. Then, following Equation (IO), we have the conditional vector field:
dV/
(0, c) =) (
/ p'()(( (k,,
(11)
dk
for k E [[k]. [k] + 1) and == k [k]. Then, following Eguation (8) the training objective of K-Flow is to learn the unconditional vector field in Equation (5) by the conditional flow matching:
CK.FION = Evo /" débu dk
<VV dk
vn vv
(12)
By examining Equation (10) closely, we observe that the vector field is naturally localized around a subset of points in the K K-amplitude space that satisfy / A2 + K2 + k2 E [[k], [k] + 1). This lo- calization means that the reconstruction at any given k primarily involves K-amplitude components within narrow frequency band around k. Compared with the flow scheme in the pixel space, the K K-amplitude in K-Flow reduces the optimization complexity by restricting the conditional vector field to be within sub-manifold for each k. This sub-manifold is of low dimensionality, allowing for more focused updates and reducing the optimization space's dimensionality at each step. We will investigate how this localized conditional vector field affects the generation path in Section B The inference computational complexity of our method is discussed in section D
We can further generalize the interpolation interval from (LA: Lk]
1) to (km: kn), where km and kn are two integers such that km < kn. This adjustment broadens the range for intermediate
amplitude components from 2 + + A3 E [[K], [k] + 1) to / 12 + hi + k2 E [kim kn). For
example, for our experiments, we partition the K-amplitude into three parts. See Section D for detailed implementations of these partitioning strategies.
3.2 EXAMPLES OF K-AMPLITUDE TRANSFORMATION
As we can see from Equation [ all K-amplitude decompositions are achieved through expansion across complete set of basis functions. However, the behavior of a K K-amplitude decomposition (transform) can vary significantly depending on the choice of basis functions. Besides the Fourier transform introduced in Section we provide two representative examples of K K--amplitude decom- position: Wavelet transformation, and PCA transformation. More details are in Section D
Wavelet Transform. Wavelet decomposition (transform) deals with data that are not only scaling- localized but also spatially localized. The scaling parameter of wavelet transform is closely related to the notion of compose a signal into approximations and details at successively finer seales. This hierarchical de- composition is achieved through set of scaling functions (Ir), and wavelet functions (0xx). which
analysis (Mallat 1989), which provides systematic way to de-

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together serve basis functions for the wavelet transformation. More precisely, in the wavelet trans- form, a signal f(t) is expressed as a sum of scaled and translated versions of these basis functions times the corresponding coefficients and d:
f(1) Σ Cko.j (1+ (k.)) 12ke
(13)
where W (1) and Vk. are the scaled and translated scaling and wavelet functions, respectively. The index which originally denotes the translation parameter, groups the basis within each fixed scaling parameter naturally. Let do. := Σ, jd.. j for > Ko and @ == 'S Cko. for = ko. then eq. [3] is just one realization of K K-amplitude decomposition. Concrete formulas for different families of wavelet bases, such as Daubechies (db), can be found in Section D
In this article, we employ the discrete version of wavelet transform (DWT) as our K-amplitude transformation F, which shares the linearity property with the Fourier transform with bounded scaling parameter k. providing a structured yet flexible means of decomposing discrete data.
Data-dependent PCA Transform. Note that Fourier and wavelet decompositions are nonpara- metric k-amplitude decompositions that are independent of data. While these transformation methods are powerful, also aim to find data-dependent decompositions that can capture common characteristic features specific to a dataset. This motivation leads to principal component analysis (PCA), technique widely used for the low-dimensional approximation of the data manifold and vision features (Izenman 2012) Chen et al. 2024). Please consult Section D for the K K-amplitude realization of PCA transform.
K-transform without the VAE's latent. Our current K K-ampltitude decomposition operates channel-wise on the given input representation (a VAE's latent). We also note that other efficient representations of images (like RGB to YCbCr with sparse discrete cosine transform Nash et al. Ning et al. (2025)) could yield a superior input space for our K-decomposition. Given that our current results built upon established pre-trained VAEs, fully integrating new transform space would necessitate training new auto-encoder models, and we see this as a very promising avenue for future investigation.
4 EXPERIMENTS
We conduct comprehensive experimental evaluation of K-Flow focusing on its technical innova- tions in k k-amplitude adapted generation. Using standard backbone architectures, we perform exten- sive experiments spanning image and molecular assembly generation tasks and k scaling guidance editing and restoration. Complete implementation details, experimental configurations, ablation studies, and scientific generation tasks provided in Section E (Algorithm) and Section D
4.1 IMAGE UNCONDITIONAL AND CONDITIONAL GENERATION
Image Unconditional Generation The first task is to generate random samples after fit- ting a target data distribution, which is typi- cally concentrated around a -iddimnnsional sub- manifold within the ambient space.
Table 1: Unconditional generation on CelehA-HQ
Model
FID.
Recall
K-Flow, Fourier-DiT DiT------//// 1.2 (Ours)
5.11
0.47
K-Flow, Wave-DiT 1/2 (Ours)
4.99
0.46
K-Flow, PCA-DIT L/2 (Ours)
5.19
0.48
Dataset and Metrics. We conduct experiments on the CelebA-HQ (Karras) 2017) dataset with
LFM. ADM [Da et al. 2023
5.82
0.42
LFM, DiT L/Z (Dto
2023
5.28
0.48
WaveDiff, DiT D2 Phung ct 2023
5.38
0.44
the resolution of 256 x 256. To evaluate the performance of our proposed method, we em- ploy two metrics: the Fréchet Inception Dis- tance (FID) (Heusel et al. (2017), which evaluates the quality by measuring the statistical similarity between generated and real images, and Recall
FM [ippmun et al
2022
7.34
LDM Rombach et al. /222
5.11
0.49
LSGM Vahdal el m.l. 2024
7.22
WaveDiil (Phung
2023
5.94
0.37
DDGAN Xiao et at. 2021
7.64
0.36
Score SDE (Song ct 2012 2022
7.23
Kynkäänniemi et al. 2019), which measures the diversity of the generated images.
Results. Table summarizes the comparison between our proposed K-Flow model and other gen-
erative models. For fair comparison, both the baseline ordinary flow matching (LFFM (Dao et al.
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) and our K-Flow flow utilize the same VAE's latent from (Rombach et al. 2022). Our K-Flow employs a lightweight MoE variant of the Diffusion Transformer (DiT L/2 (Peebles & Xie 2023
as the backbone model, while LFM uses the standard DiT L/2 architecture. We can observe that (1) K-Flow achieves the best performance in FID, especially w/ the db6-based wavelet K-Flow. (2) Al-
though the latent diffusion model (Rombach et al. 2022) gets the highest score in Recall (diversity),
the Fourier and PCA-based K-Flow is comparable with the ordinary latent flow matching.
Image Class-condittonal Generation Then we explore how K-amplitude decomposition be- haves when the generation path is conditioned on class labels, where the class label (e.g., dog, cat, fish, etc) delegates the low-scaling informa- tion of each image. This investigation could potentially pave the way for multi-scaling con- trol, where different scaling components are in- fluenced by specific caption information. We list the detailed class- canss-oonditional generation algo- rithm in [e
Table 2: Clndsicon generation on ImageNet.
Model
FTD
CDR,
Recall
K-Flow, Wave-DIT (Ours)
17.8
+ =cfg=1.5
4.49
0.44
K-Plow, PCA-DiT 1/2.,fg--1 K.Dlow Fourier DIT L// (Ours)
4.19
0.43
13.4
0.57
+ cffg1115
2.73
1.49
0.45
D/2
+ cfg=1.5
2.85
3:25
0.42
LDM-8 (Rombach al. LDM-8-D
202
15.51
0.63
7.76
0.35
DIT-B/2 Peebles & X
2025
43,47
VAR-d16 TTan 2024
(cfg=2.0)
3.30
Dataset and Metric. We use ImageNet with res- olution 256 256 as the middle-size conditional
FlowAR- Keen et a.
212
(clg=2.0)
3.17
0.52
FlowAR-L Rea
(cfg=2.4)
generation dataset (Deng et al. (009).. Beyond evaluating the unconditional FID for the ImageNet dataset, we are also interested in studying how the class control interacts with scaling generation in quantitative manner.
The results are presented in T Our primary focus for the FID metric the classifier- free guidance inference method applied to flow matching models. The data indicates that K-Flow achieves results comparable to LFM. In terms of the recall metric, which assesses the diversity of the generated distribution, our model outperforms the standard LFM. This improvement may be attributed to the fact that the inference path of K-Flow includes a greater number of dimensions during the low-scaling period, discussed in Section B. We extend our comparison include two modern multi-scale auto-regressive models, VAR (Tian et al. 2024) and FlowAR (Ren et al.) 2024), with architectural and performance details provided in Table S2 Among models of a com- parable scale, our K-Flow (Fourier) achieves a superior FID of 2.72. demonstrating the efficacy of our framework. The oftte-- FlowAR-L (FID 1.90) serves as an upper-bound reference, as it utilizes a significantly larger model and different VAE, placing it in different comparison category.
Results.
4.2 IMAGE CONTROLLABLE CLASS-CONDITIONAL GENERATION
The latent flow matching model can implicitly learn low- and high-resolution features (Dao al. 2023), but the boundary between each resolution is vague, and we cannot explicitly determine which timestep in the inference process corresponds to specific resolution or frequency. In comparison, our proposed K-Flow enables finer finer-graines grained controllable generation. As demonstrated in Figures and K-Flow effectively preserves high-frequency details even when class conditions are omitted during the last 70% of scaling steps, whereas ordinary latent flow exhibits significant blurring. To quantitatively validate this observation, we analyze the conditional discrimination ratio (CDR, formally defined in Section F.. From Table 2] our model maintains CDR close to one, indicating robust performance regardless of high-scale condition omission, while conventional LFM shows significantly higher CDR, suggesting performance degradation. The class-dropping experiment provides quantitative evidence that K-Flow learns a disentangled generative path by encoding high-level semanties into low K-amplitude bands, highlighting its potential for improved computational efficiency by allowing control conditions to be omitted during later synthesis stages.
4.3
IMAGE
GENERATION AND RESTORATION
Our method guarantees that the generation path is disentangled with respect to & (check Section This allows us to control initial noise each scaling level (see Algorithm 2) enabling unsuper-
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Undrapped
diat
Ohaped
(a) Controllable class-conditional generation.
(b) Scaling-controllabb generation (low scaling).
Figure 3: Pipeline of two ablation studies on controllable generation.
K-Flow (ours)
LFM
Undrop
Umdrop
Drop
Drop
Undrop
Undrop
Drop
Drop
Undrop
Undrop
Drop
Drop
Figure 4: Results of controllable class-conditional generation. 'Drop' means we drop the class conditions during the last 70% scaling steps, while 'undrop" means we keep the condition all the time.
vised editing of different scaling components. We also conduct ablation studies on Preserving Low Scaling, Modifying High Scaling, Please check Section F for more details and visualizations.
Preserving High Scaling, modifying Low Scaling. This maaling-controllbbe generation pipeline illustrated in Figure 3b It involves sampling multiple images while ensuring that the noise in the image gen-
high-scaling components remains consistent across all samples. In scaling-controllate eration, the goal maintain consistency in the high-scaling details while allowing variations the low-scaling context among the generated images, thus this allows K-Flow to achieve unsupervised steerability in finetuning free manner.
The results on CelebA are presented in Figure 5 where we apply a pretrained Daubechies wavelet (db6-based) K-Flow. can observed that facial details, such as eyes, smiles, noses, and eye- brows, remain consistent within each group of images. In contrast, the low-scaling components, including background, gender, age, and hairstyle, vary across the images within the same group- These qualitative results demonstrate how frequency bands naturally correspond to semantic fea- tures facial details persist in high-frequency components while attributes like global background and overall appearance vary in low-frequency components. When applying the same editing proto- col to conventional LFM, the results show no such organized frequency-semant semantic correspondence (Figure SIO), highlighting the advantage of K-Flow.
Image Restoration. While traditional semantic metrics cannot directly assess unsupervised frequency-based editing, we quantitatively validate our scaling-aware generation path via image restoration tasks, where frequency-specifie changes are objectively measurable. This includes super- resolution and deblurring. From Table S6 in Appendix, K-Flow achieves state-of-the-an perfor- mance in terms of PSNR and SSIM metrics on the CelebA dataset. This demonstrates that frequency-done domain formulation enables accurate high high-frequency reconstruction. Detailed experi- mental settings and algorithms provided in Section F

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Figure 5: Results of controlllle generation: Each column presents an image pair that shares high-
frequency components while exhibiting distinct low-frequency characteristics.
CONCLUSION
In this paper, we introduce K-Flow Matching (K-Flow), an efficient flow-matching model that flows along the K K-amplitude for generative modeling. K-Flow naturally generalizes the multi-scales of data (e.g., multi-resolution or frequencies in images) to multi-scales the K K-amplitude space.
Limitations and Future Directions. As we have verified the effectiveness of K-Flow exclusively on image generation tasks, moving forward, three promising directions are worth exploring. (1) Multi- modal generation that includes tasks such as large-scale data generation guided by dense captions. This is an extension our class-dropping experiment, which could better showcase the efficient stecrability of K-Flow by aligning images with natural language inputs. (2) We outline six proper- ties of K-Flow in Section e.g., the amplitude naturally corresponds to energy. While Section briefly discusses how energy is represented in K-Flow, this aspect has not been explored in depth. We believe that such energy term holds potential for integration with the utility of energy-based models in future work. (3) Unifying understanding and generation: Our method demonstrates that generation can be performed in a scale-aware manner, particularly by following a coarse-to-fine fre- quency ordering (i.c., from low to high frequencies). From representation learning perspective, low-frequency components naturally encode global semantic features that are closely aligned with those learned in visual understanding tasks. This observation suggests that K-Flow holds potential for bridging the gap between visual understanding and generation, by incorporating representations pre-trained on understanding tasks into the generative process. We regard this as compelling rection for future exploration toward unified visual modeling.
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ETHICS STATEMENT
This research adheres fully to the ICLR Code of Ethics. It does not involve human subjects, personal information, or sensitive data. All datasets and code used released comply with their respective licenses and terms of use. The contributions of this work are methodological and foundational, raising no concerns related to fairness, privacy, security, or potential misuse.
ACKNOWLEDGEMENTS
We thank the reviewers for their valuable feedback and suggestions that helped improve this work.
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THE USE OF LARGE LANGUAGE MODELS
We employ LLM refine the language and edit the draft of this paper, including:
Correcting grammatical errors, punctuation, and spelling.
Improving sentence structure to enhance clarity, flow, and readability.
Suggesting alternative phrasing for more precise and professional academic expression.
All modifications suggested by the LLM were critically reviewed, vetted, and approved by the thors. The final text accurately reflects our own ideas, arguments, and research findings.
A PROPERTIES OF K-FLOW
(a) K-Flow provides first-principle way to organize the scaling k. Unlike perception-base computer vision tasks, which often favor certain scaling (frequency) bands, K-amplitude based generative model strives for an optimal organization of all scalings to ensure that the final generated sample of high fidelity. By constructing K K-amplitude scaling-based vector fields, the integrated flow naturally incorporates all scaling information, and the conditional flow matching training ob- jective provides a perfect trade-off of excuracy-efficieme demonstrate how different discretizations of K-Flow with related works, highlighting the connec- tions and integrations with existing methods in the field.
inside localized scalings. We will also
(b) K-Flow enables multi-scale modeling in the K amppliuude space. Compared to the original data space, such as the pixel space in images, the K K-amplitude space provides a more natural perspective for defining and analyzing multi-scale information, namely, K -ampplit decomposition empowers K-Flow for effective multi-scale modeling. By decomposing the feature representation into multiple scaling components in the K K-amplitude space, K-Flow associates each scaling with amplitude. Higher values of K --amplitude correspond to higher- freuueccy information, capturing fine- fine-grained details, while lower values encode fower-frequency information, representing more coarse- coarse-grained features. Let us take the image for illustration. Images inherently exhibit hierarchical structure, with information distributed across various resolution levels. Low-resolution components capture global shapes and background information, while high-resolution components encode fine details like textures, often sparse and localized. By projecting these components into the K-amplitude space, K-Flow captures such hierarchical information effectively and naturally, enabling precise modeling of the interplay between scales.
(c) K-Flow supports well-defined scale along with energy. The amplitude is also used to reflect the energy level at each scale of the data. In physics, it is proportional to the square of the amplitude. In comparison, for the modeling on the original data space, though we inject application-spectn inductive bias, such as multiple pixel resolutions for images, they do not possess a natural energy concept.
(d) K-Flow interprets scaling as time. From elucidating the design space of the traditional flow matching perspective, K-Flow re-defines the artificial time variable (or the signal-to-noise ratio vari- able proposed in (Kingma et al., (022)) as the ordering index of frequency space. In this context, the artificial time variable effectively controls the traversal through different levels of a general notion of frequency decompositions, scaling each frequency component appropriately. This perspective aligns with the concept of renormalization in physical systems, where behavior across scales is sys- tematically related.
(e) K-Flow supports the fusion of intra-scaling and inter-scaling modeling. K-Flow flows across scaling as time, and namely, K-Flow naturally merges the intra- and inter-scaling during the flow process. Thus the key module turns to the smooth interpolant, as will introduced in Section 3 This is in comparison with existing works on multi-modal modeling (Burt & Adelson 1987 Tian ct al. 2024] Atzmonn et al. 2024), where the special design of the intra-scaling and inter-scaling is required.
(f) K-Flow supports explicit steerability. The flow process across scales enables K-Flow to control the information learned at various hierarchical levels. This, in turn, allows finer-grained control of the generative modeling, facilitating more precise and customizable outputs. By understanding and leveraging K-Flow's steerability, its utility can be significantly enhanced across diverse domains,
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including Artificial Intelligence Generated Content (AIGC), AI-driven scientific discovery, and the safe, responsible development of AI technologies.
Inteligence-GGeneaatee
B DISCUSSION
B. FROM CONDITIONAL TO UNCONDITIONAL PATH IN K-FLOW
In Section 3 our frequency-localized path is defined at the conditional level ( (cb and it
only related to the unconditional vector field (v% (Vx, 0) in eq. (12) through the equivalence of conditional flow matching and unconditional flow matching at the loss level (Lipman et al. (022). In this section, we try to study the splitting property of the unconditional K K-amplitude vector field.
By the -aamllitude decomposition, the transformed data probability Pdata satisfies the telescoping property:
Pdato (pkk))p(k///e) (oo))..-((knnn Kmar 1...., ko);
(14)
with ko and kynn denoting the lowest and highest scaling. Then, according the definition of our proposed K-Flow kx the interpolated probability scaling step t also localized:
p(()) = p(ko) pa (
(o)).((8:
1) ....... (kmas (Riuez 1, 1,.... ko):
(15)
+ +1). Combining Equa- tion (15), the localization property of the bump function, and Lemma of (Zheng et al.) 2023). the
where P. denotes the distribution of the initial noise and [[A], [k]
unconditional vector field has an explicit form: by are hyper-paramelers determined by the bump function we choose.
(V)))) a, 1xx + be log Pt (V&), where d, and
200
Low Scaling
Middle Scaling
Average Norm
150
High Scaling
100
50
Channel
Channel 2
Channel
Channel
Supplementary Figure S1: On the low-scaling hypothesis. The graph illustrates the relative norm dis- tribution for each scaling component as defined by wavelet decomposition in the latent space. can be observed that the low-scaling component exhibits a significantly higher norm (energy), nearly twice that of the high-scaling component.
Noise Splitting. A key characteristic of flow models is their deterministic nature after the initial noise sampling. Specifically, once the initial noise is sampled, the flow follows a fixed path to generate the final data sample. According to Equation (15), during scaling step t: (1) the scaling components below [k] remain unchanged; (2) the scaling components above [k] remain unchanged; (3) The distribution of higher scaling components maintains the same characteristics as their initial noise distribution.
By these observations. we now investigate how segmented initial noise in the K-Flow space influ- ences the final output of the K-Flow flow. Suppose we discretize scaling parameter & into two parts: F = (srmm (k), (higg }. When flowing along the low-scaling component, the vector field eg can be re-expressed in conditional form:
Up (v²) = v (k),
(16)
where constant represents the (static) initial noise for the high-scaling part. This noise-conditioned property the k-amplitude domain leads us to explore whether fixing the high-scaling noise and altering the low-scaling noise allows for unsupervised editing of relative low-scaling semanties in image. Indeed, we observed this phenomenon, the qualitative results will be discussed Section
From Figure 5 we observe that targeted common high-scaling initial noise guides our K-Flow flow toward generating human faces with similar detail but varying low-level content. See the experiment section for more detailed analysis.
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100
Projection Eviluaion
80
LFM
K-Flow
0.5
1.0
Scaling Step
Supplementary Figure S2: Projection Error Comparison with Different flow Models. The graph il- lustrates the PCA projection errors of two trained models throughout the entire inference process, with distinct segments marked by dashed lines. The red and blue lines represent the original latent flow matching (LFM) and the K-Flow with amplitude components, respectively. The projection error is quantified by the recon- struction error each generation step from the PCA compression, using the first two principal components. Owing to the scaling-aware nature of our flow, the low-amplitude portion (the initial part of the curve) resides in relatively high-simensional space, resulting in higher projection errors for the two-dimensional PCA pro- jection.
B.2 THE EFFECT SCALING STEP i FOR IMAGE RECONSTRUCTION
K-Flow's ability to leverage the low-dimensional structure of data is primarily enabled by its K- Flow localization property. This enables a strategic path through low-dimensional spaces, which can be directly compared with the generation path of conventional flow models. our model, this path incorporates explicit frequency hierarchy, which hypothesizes that the www-freuuency com- ponents concentrated in the earlier stages of the model may share more dimensions in common, particularly from semantic perspective, than the high-frequency components positioned later in the generative process. Conversely, ordinary flow model may exhibit a more uniform distribution of dimensionality across the entire generative path.
Motivated by this hypothesis, we conduct a case study using PCA to approximate the dimension kmaz of the generation trajectory { k= As illustrated in Figure S2 we measure how closely the dimension of the generation path aligns with -ooddimensional subspace spanned by the first two components of the model's ''s PCA decomposition, denoted by ) kmuz Inspired by Zhou et al,
{
2024,, the reconstruction ratio is defined by 1 V| V / In other words, a higher value of the reconstruction ratio indicates that the model's dimension is closer to two. Therefore, the trend of the eиrоr curve with respect to the scaling parameter & reveals distinct separation in the effective dimension between low- and high-scaling components. Evidently, the low-scaling segments display more semantic consistency and thus, occupy larger dimensional space, whereas the high-scaling segments converge to a more confined or lower-dimensional structure.
It is important to note that this exploration into the dimensionality of generative paths is practi- cally meaningful. Previous study (Zhou et al., 2024) has shown that the effectiveness of distilling generative model with fewer steps from pre-trained diffusion model theoretically depends on the model's dimensionality at each step. as informed by the high-dimensional Mean Value Theo- rem. The observations from Figure S2 provide empirical support for this concept. Specifically, the ability of K-Flow to maintain a lower-dimensional structure in high-scaling components suggests promising approach for fast sampling distillation methods.
B.3 RELATED WORK DISCUSSION
The field of generative modeling has seen significant advancements in recent years, driven by a
variety of frameworks, including adversarial generative networks (GAN) (Goodfellow al. 2014).
variable autoencoders (VAE) (Kingmal 2013), and normalizing flows (Papamakarios et all 202
In this work, focus continuous normalizing flow generative models (Chen al. 2018). with
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Data Space
IntenScclle Modeling
K-Amplitude Space
Data Space
Supplementary Figure S3: Comparison of multi-scale modeling: pixel data space K-Amplitude space.
particular emphasis on the conditional flow matching training scheme, which originates from the denoising score matching training framework (Vincent 2011).
Both diffusion models and continuous flow matching models aim to lower the complexity directly optimizing the log- likeliikiihoo of data by introducing additional stochastic path. However, as proved (Lavenant & Santambrogio, (202), the canonical path for diffusion models and rectified flows is not optimal. This realization motivates introduction of frequency decomposition key design element in generative models.
By breaking down the formula of our K-Flow vector field with respect to the scaling parameter k; we can summarize three successful factors as general principles for scaling modeling.
A good K-amplitude decomposition certain scaling bands. For generative tasks, is crucial that all K-Flow bands are effectively modeled ensure the generation of high-quality, controllable outputs. In addition, the computational resources required may vary between different scales, thus necessitating careful consideration of resource allocation.
leverage the problem's inherent biases towards
Modeling within each scaling component, which is formulated in our K-Flow-localized vector fields.
Modeling bridges along different scalings, which is achieved through our flow ODE and the (time) K-Flow embedding block for the U-Net or DIT architecture.
This approach to inter- and intra-modeling for K- K-amplitude is also applicable to scenarios emphasiz- ing certain frequencies or scalings. For instance, (Li et al.) 2024a) enhanced oscillatory motion con- trol in video generation by discarding the high-frequency component of the Fourier decomposition. As discussed in Section [] the scaling parameter of spatially localized wavelet (multi-resolution) decomposition is closely linked to image resolution. Notable contributions in this domain include (Atzmon et al. (222)) and (Lei et al.) 2023), which introduced a multi-stage resolution for fine- grained editing, and (Jin al. 2024), which concentrated efficient video generation. We provide a systematic review of frequency- or scaling- based generative approaches in Section C highlighting key developments this direction.
In related research on auto-regressive modeling, (Mattar et al. 2024) presented wavelets as an effec- tive auto-regressive unit, while (Tian al., 2024) focused the scale key element for image auto-regression. A significant example is (Phung et al.) 2023), which transitioned the latent space from pixel to wavelet space for generative models using wavelet diffusion. However, their method employed the same conditional noising schedule for score matching as traditional diffusion models. In contrast to their approach, our proposed K-Flow integrates wavelet decomposition as multi- channel module within the neural network architecture for training diffusion models. Additionally, our work extends the notion of wavelet space to the more general K-amplitude space.
We also want to highlight another research line that has recently caught the attention: the auto- regressive modeling over the pixel space for image generation. One classic work is VAR Tian et al. (2024). It introduces a hierarchical down and up sample paradigm that models images in coarse- to-fine manner across multiple resolutions and models the data distribution in an auto-regressive manner. In contrast, our proposed K-Flow integrates the flow paradigm for density estimation and
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leverages the K-amplitude space stronger inductive bias, as illustrated in Figure S3 Another related work is the auto-regressivv flow model proposed in (Ren et al. 2024). that implements con- ditional flow matching sequentially across scales. Although (Ren et al. 2024 shares some common terminology with our work (e-g., scales, flow matching), K-Flow provides significant advantages through its unified flow process, almost architecture-agnostie design, and theoretically grounded fre- quency domain framework. On the practical implementation side, Unlike (Ren et al.) (024), which requires separate flow matching for each scale and relies on specific architectures (autoregressive transformers that treat scaling as conditional input), our approach implements a single coherent flow that connects all frequency scales during inference while maintaining architecture flexibility.
Summary. In summary, K-Flow is a more general framework, with its three key factors potentially benefiting generation-related tasks like super-resolution and multi-resolution editing. For example, (Liu et al.) utilized a learnable Fourier transform to construct harmonic module in the bottleneck layer of autoencoder. We provide a comprehensive list of related works in Section C
B.4 CONNECTING K-FLOW WITH SSL REPRESENTATION AND GENERATION
From the above discussion, have seen how pretrained vision models leverage the sparsity and locality of natural data in various K K-amllitude domains for perception and generation- based tasks. In the realm of unsupervised learning, (Liu et al. 2022a 2024a] Chen al. 2024 explore whether generative-baeed representations, particularly those derived from denoising diffusion models, can achieve parity with contrastive-baser representation learning methods for downstream tasks. A key observation from their findings Chen et al.] (2024), which aligns with our approach of employing K K-amplitude decomposition (the PCA instance), is the revelation that the most powerful represen- tations are obtained through denoising within a latent space, such as the compressed PCA space. Another merit of PCA is that denoising along the PCA directions can achieve faster convergence for denoising, which is revealed in (Du et al. 2023).
To transition from unsupervised representation learning to real data generation, incorporating all K- amplitude scalings is essential. Rather than compressing amplifying specific scaling bandwidths, generative tasks require novel organization or ordering of all frequencies. Besides our flow-base frequency generation approach, (Tian et al. 2024) connects different scales (which can be inter- preted as the wavelet K K-amplitudes) using residual connections with an auto-regressive training objective. Residual connections, as discretization of ordinary differential equations (ODEs) pro- posed in (017)) suggest that (Tian et al.) 2024)'s 's approach can be seen as a special discretization of our K-Flow with a flexible flow matching training objective.
Supplementary Table S1: Comparison among PCA, contrastive, and generative SSL.
Basis Learning
Reconstruction Learning
PCA SSL Contrastive SSL Generative SSL
non-paraaeteriize, Determined By Data
Parameterized
Parameterized Parameterized
N/A Parameterized
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C RELATED WORK ON FREQUENCY, AND MULTI-SCALING
There have been multiple research lines on studying generative modeling, especially in terms of multi-scale modeling. In this work, would like summarize them the following three venues.
C.I MULTI-SCALE IN PIXEL RESOLUTION, FLOW AND DIFFUSION
Laplacian Pyramid and Laplacian Operator. In mathematics, the Laplacian operator computes the second derivative of a function, emphasizing regions with significant intensity changes, such edges nigh-frequency details. Similarly, the Laplacian Pyramid (Burt & Adelson 1987 de- composes an image into multiple scales, extracting the low-frequency components (smooth regions) through downsampling. The high-frequency details, such edges and textures, are modeled as the residuals between adjacent resolution layers. The primary objective of the Laplacian Pyramid is to represent these residuals across scales in a hierarchical fashion.
LAPGAN (Laplacian Generative Adversarial Networks) (Denton et al. 2015) adopts the Lapla- cian pyramid idea into the generative adversarial network (GAN) framework (Goodfellow et al., 2014]. By focusing on learning residuals between successive levels of resolution, it effectively gen- erates high-quality super-resolution images.
SR3 (Super-Resolution via Repeated Refinement) (Saharia et al., 2022) leverages DDPM (De- noising Diffusion Probabilistic Models) (Ho et al.) 2020) and DSM (Denoising Score Match- ing) (Vincent 2011 [oog & Ermon 2019) for high-resolution image generation. Specifically, SR3 enhances low-resolution images to high-resolution by utilizing multiple cascaded conditional diffu- sion models. In this framework, the low-resolution images serve conditions, and the model's aim is to predict the corresponding high-resolution images as outputs.
PDDPM (Pyramidal Denoising Diffusion Probabilistic Models) (Ryu & Ye 2022) is follow-up work of SR3, and it improves the model by only modeling one score network. The key attribute to enable this is by adding the fractional position of each pixel to the score network, and such fractional position information can be naturally generalized to different resolutions.
F-DM (Gu et al. 2022 is developed concurrently with PDDPM and shares the approach of utilizing only one diffusion model. It distinguishes itself by explicitly applying a sequence of transformations to the data and emphasizing resoouiion-agnostic signal-to-noise ratio within its diffusion model design.
Edify Image (Atzmon et al. 2024) is state-of-the-art model capable of generating photorealistic, high-resolution images from textual prompts (Atzmon et al. 2024). operates as a cascaded pixel- space diffusion model. To enhance its functionality, Edify Image employs downsampling process that extracts low-frequency components and creates three distinct resolution levels, ranging from low high frequency, with the original image representing the highest frequency level. Another key innovation of Edify Image is its meticulously crafted training and sampling strategies at different resolutions, utilizing attenuated noise schedules.
C.2 MULTI-SCALE IN PIXEL RESOLUTION, VAE AND AR
VQ-VAE2 (Vector Quantized VAE 2) Razavi al. 2019) enforces two-layer hierarchical struc- ture. where the top layer captures global features such as object shapes and geometry, while the bottom layer focuses on local details like texture. It models data density within the variational autoencoder (VAR) Beanework(Kingna 2013) and incorporates an autoregressive (AR) module to enhance the prior for improved generative performance.
RQ-VAE RRsidual-Quuntiizd VAE) (Lee et al.) 2022) integrates recursive quantization into the VAE framework. It constructs a representation by aggregating information across D layers, where the first layer provides a code embedding closely aligned with the encoded representation, and each subsequent layer refines this by reducing the quantization error from the previous layer. By stacking D layers, the accumulated quantization error minimized, enabling RQ-VAE to offer coarse-to- fine-grained approach to modeling. For modeling, the general pipeline follows the VAE framework, while each latent code is decomposed into D layers and is predicted in autoregressive manner.
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VAR (Visual AutoRegressvve) (Tian al. 2024 introduces novel paradigm for density estimation by decomposing images into multiple resolutions across various scales. This approach is inspired by the hierarchical nature of human perception, where images are interpreted progressively from global structures to finer details. Leveraging this concept, VAR models the entire image in a coarse-to-fine manner, adhering to the principles of multi-scale hierarchical representation.
C.3 MULTI-SCALE IN FREQUENCY, AR, VAE, AND DIFFUSION
WaveDiff (Wavelet Diffusion) (Phung al. 2023) leverages the discrete wavelet transform to shift the entire diffusion process into the wavelet spectrum. primary objective to reduce model complexity by operating in the transformed spectrum space instead of the pixel domain.
PiToMe (Protect Informative Tokens before Merging) (Tran et al., 2024) is a token merging method designed to balance efficiency and information retention. PiToMe identifies large clusters of similar tokens as high-energe regions, making them suitable candidates for merging, while smaller, more unique, and isolated clusters treated as low-energy and preserved. By interpreting attention over sequences as a fully connected graph of tokens, PiToMe leverages spectral graph theory to demonstrate its ability to preserve critical information.
WF-VAE (Wavelet Flow VAE) (i et al. 2024b) is a parallel work that injects the Wavelet transform into the backbone model of the VAE framework for extracting the multi-scale pyramidal features. We need to emphasize that WF-VAE introduces frequency decomposition as an inductive bias into the backbone model to simulate energy flow. In contrast, our K-Flow retains the backbone archi- tecture and instead injects the K-amplitude as the realm for energy flow. In other words, K-Flow incorporates the multi-scale concept through the time domain.
SIT (Spectral Image Tokenizer) (Esteves et al. 2024) is parallel work to ours that processes the spectral coefficients of input patches (image tokens) obtained through a discrete wavelet transform. Motivated by the spectral properties of natural images, SIT focuses on effectively capturing the high- frequency components
images. Furthermore, introduces a scale-wise attention mechanism, referred to scale-causal self-attention, which is designed to improve the model's expressiveness across multiple scales.
dCTdiff (DCT Diffusion) (Ning et al.) 2025) explores generative modeling in the Discrete Co- sine Transform (DCT) space, demonstrating that operating in the DCT domain can yield improved generation quality. While DCTdiff shares our motivation of leveraging frequency-domain transfor- mations, it focuses specifically on the DCT representation as a fixed transform. In contrast, our K K-amplitude framework provides more general decomposition that encompasses DCT, wavelets, and PCA, offering greater flexibility in choosing the appropriate frequency representation for differ- applications.
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D METHOD DETAILS
D.1 FOURIER TRANSFORM AS A K-AMPLITUDE DECOMPOSITION
We have shown how to build the K-Amplitude through the Fourier space Section 2.2 In the discrete setting, the Fourier transform realized by basis functions of the form W e - km where N is the length of the sequential data. An effective K-amplitude decomposition exploits this structure by aligning with the inherent hierarchical structure of the data manifold. For example, most of the energy or amplitudes are concentrated in the low-scaling range, the generative capability of the flow can be enhanced by allocating more steps or resources to these low frequencies (this hy- pothesis is even true the latent space, as it's demonstrated fig. ST). Conversely, fewer steps can be allocated to high frequencies that carry minimal mass or information. For the Fourier transform, this tendency is evident in the analysis of natural images, which often exhibit the celebrated 1/f spectrum phenomenon (Weiss & Freeman 2007). This phenomenon suggests that energy dimin- ishes with increasing scaling parameter, meaning that low-scaling components hold the majority of the signal's s information content.
D.2 PCA TRANSFORM AS A K-AMPLITUDE DECOMPOSITION
From the K -ampliitdde perspective, PCA is an eigen-decomposition obtained by the data covariance matrix. The covariance matrix is given by:
C X n
centered Xenttredd
where Xeentered X - X is the centered data matrix. In this context, the principal components reveal the relative importance of each transformed direction. To translate PCA into a K K-amplitude decomposition, we define the scaling parameter the relative order of the principal components. For implementation, we utilize the eigenvalue decomposition of C for PCA, and the eigenvalues in their descending ordering define the scaling parameter
D.3 DWT TRANSFORM AS A K-AMPLITUDE DECOMPOSITION
The Discrete Wavelet Transform (DWT) (Akansu & Haddad 1992) is utilized to decompose a sig- nal at multiple scales, capturing both time and frequency characteristics. It involves scaling and translating wavelets.
The DWT decomposes the input signal into approximation and detail coefficients:
Given a discrete signal æ/n] (expressed by a knite-dimension vector), use the scaling function @(t) and wavelet function (t) to generate coefficients:
c+bi+ Σ r ria] Φx In),
d&Li Σ x[n] [n))
Here, cx]i are the approximation coefficients at scale k, and dgL are the detail coefficients at scale k. Comparing with our definition K-Flow decomposition, A: just a discrete scaling parameter.
The inverse transform then reconstructs the original signal from the coefficients:
Σ
(alloo + Σ (k
Recursive Relationship between different Scales (k) Different levels of decomposition recur- sively related:
& single level decomposition results approximation coefficients e and detail coefficients dj;
2. A 2: A two-level decomposition first produces coefficients с₁ and d1. Then, the ap- proximation coefficients ç are further decomposed into a second level of approximation coefficients c2 and detail coefficients d2.
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For k = 2. the decomposition looks like: z|n]] + (o2, d1) where d1 represents the high- frequency components (level I detail coefficients) and e1 is the low-frequency component (level approximation coefficients). Further decomposing e yields C2 (level 2 approximation coefficients) and d2 (level detail coefficients). This recursive relationship illustrates why we can effectively take finite maximum scaling parameter kmaz and still own inverse transform.
Practical Design choice. In this paper's experiments, especially the Wavelet version of K-Flow flow, we take the kmaz to be one or two. One means decomposing the data into two scales, and two means decomposing the data into three scales.
Pre-conditioning the data based on scaling As illustrated in fig. ST the energy distribution exhibits significant heterogeneity across different frequency bands, which consequently leads to non-uniform vector field norms in our localized K-Flow. To address this training instability, two approaches can be considered: First, following the methodology proposed in conventional diffusion models (Karras) al. 2022). we could incorporate input-output preconditioning modules into the neural architecture. However, this approach necessitates modifications to the backbone network structure, potentially affecting model compatibility and transfer learning capabilities.
In this paper, we propose a more flexible solution through component-wise normalization of the initial noise in the frequency domain. Specifically, after transforming the initial noise to the K- amplitude space, we normalize each discretized scaling component independently to have unit vari- ance (var=1). This normalization is applied only to the noise, while the data remains unchanged. This approach effectively stabilizes the training dynamics by ensuring uniform noise variance across different frequency bands. More importantly, by normalizing the noise variance to unity for each scaling component, the denoising scale of the backbone model maintains consistent magnitude across different scales, which leads to more stable optimization.
D.4 IMPLEMENTATION DETAILS OF K-FLOW VECTOR FIELD
K-Flow is built upon a lightweight MoE (Mixture of Experts) variant of the Diffusion Transformer (DiT) (Peebles& & Xie 2023), where the experts are selected based on k scales. More precisely, the integration of our method involves two key architectural modifications: (1) replacing the conven- tional time-embedding module with a ---amplitue--mbeddeee
module, where the temporal input is substituted by the scaling parameter k; (2) replacing the standard feedforward MLP modules with frequency-aware MoE modules that route tokens to different experts based on their K K-amplitude components. This design enables direct incorporation of scaling information and allows the model to process different frequency bands through specialized expert networks.
For practical implementation, we provide several variants of bump functions in this subsection to facilitate exploration of the design space, with the complete training algorithm detailed Algo- rithm I For additional insights on the K- K-amplitude localization property and its implications for model efficiency, we refer readers to Section D.5
Despite this almost model-agnostic nature, the unique K amplitud localization property of Equation (TT) offers an opportunity to design more efficient models. For instance, consider points that lie outside the support of function T. VAz+A + + In these regions, their derivative re-
Remarks.
mains zero, indicating that they do not contribute to the optimization process for the corresponding focus computational efforts solely on the val- By doing so, the values outside this region can be treated as static conditions, providing a fixed context.
scaling band. This selective activation allows
ues within the support of the indicator function, II +
Scaling Discretization. In the main text, assume, by default, that the scaling parameter A takes Thus, the differentiable vector field Uy for continuous is defined by interpolating between [k] and [k] + 1.
integer values: E {0, ... }max)-
We now extend this concept to a more general setting where k may take a limited set of integer values within the range from 0 to Amax- Suppose km and kn represent two specific integer values for We demonstrate how to extend & continuously within the connected interval [km k,). Let
t = k km- The differentiable version of dg is then expressed as:
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km + =F=1
/a3+eerAtee ky, + (1-1Vg=R2+
(etRT+EE (u(l)- F(o) Kiys hin + +1 (1-1(()) o)
(17)
where (1l) is a bump function fulfilling pd0) = (u(ki - km) = and v' - (kin - km)-
Replacing the Fourier transform with the general K K-amplitude decomposition, the K-Flow is ex- pressed in its general form as follows:
Tkcckn F(o)(K) + (I-Ive2A.) e IkElhm. (1(()) (((()))(())b)) + x(()) ()) )
Vem =
where (k) is defined the main text.
Experimental Implementation. In this paper's experiments, particularly in the Fourier and PCA versions of the K-Flow flow, we restrict the discrete values of k to {0. kan kmax J. with kmax deter- mined by resolution. We then extend continuously using Equation
Bump Function. We propose two types of bump functions: 1. Hard bump: 2. Soft bump. The hard bump function μ [0. 1] > R+ satisfies the specific endpoint properties:
p(0) = ((1)
and μ² -1(
(18)
Inspired by spline functions, such bump functions can be constructed using polynomials. For exam- ple, quartic form used in our experiments is given by:
ptt) 3t + 2t
(19)
For more examples, readers explore modifications of the connection funetions used in Meyer wavelets (Meyer 1992).
In this paper, utilize hard bump functions for constructing K-Flow flows with scaling discretiza- tion exceeding one component.
Soft Localization with Soft Bump Function. Consider that the scaling parameter is discretized to take values in an increasing sequence [ki H Consequently, the continuous & lies the interval k [Ko; k.] Define
4/ = +22
These vi form a partition of unity for the K-Amplitde basis. The derivative of the soft bump function 1e is defined for each scaling component th (a.k.a. frequency band), expressed as:
(4; a, b) = c. (1-111(( (²)=)))" 0,
a,l b, if l - a,l 2 b.
(20)
where K and e is the normalization constant ensuring that the integral of the function over its compact support is Note that hyper-parameter b < k, Ko dictates the width support region of the bump, while the degree measures the sharpness of the bump. We retain and n as hyperparameters. The bump function (k: is then obtained by integrating (k))),, which is also polynomial function.
is evident that (4(k) satisfies:
mu(ko) and p4 (k;(k,))) 1.
Finally, conditioned sampled noise the modified soft K-Flow flow time E [0, k. ku] is expressed as:
V Re F
Σ Wh Khys hz) pilko + t) Flo)(ki ky) E (ka kz) (ki + f)) -)))-e
(21)
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Supplementary Table S2: Comparison of model parameters.
Model
#Params (M)
VAR-B/16
310
FlowAR-I.
590
FlowAR-B (Our Impl.)
300
K-Flow (Fourier) (Ours)
450
Through the application of this formula and a family of soft bump functions {14), we can also implement algorithm [ In comparison to the hard bump functions, K-Flow constructed with soft bump functions assigns varying weights to each scale according to the scaling parameter k. Unlike hard bump functions which strictly set other scales zero for each stage of soft bump functions provide a more gradual transition, allowing for multiple frequencies to flow concurrently, and the relative weights are determined by the current scaling parameter k:
Comments on Haar and Meyer Wavelet K-amplitude. One type of wavelet that offers both frequency and spatial localization is the Meyer wavelet. The Meyer wavelet originally defined in the Fourier frequency domain, making it ideal for smooth frequency transitions.
The ID Meyer wavelet (t) and its scaling function @it) are defined via their Fourier transforms, (w) and d(w), respectively. The Meyer wavelet is constructed to ensure that the wavelet transform will partition the frequeney domain into octave bands.
The Fourier transform of the scaling function ((w) is defined as:
if [
I
2m 3
)(w)
cos
3/m1 2T
1))
if 2m
|W|
S
4x
3
(22)
if [
>
4m 3
where v(t) is a smooth function defined as:
if 0,
((t)
if0<t<] 1.
(23)
The Fourier transform of the Meyer wavelet ((w) then defined as:
((w) sin
3/wl 2x
1))
if 2x < |
3
(24)
otherwise.
In other words, Meyer transformation can be seen as the Fourier transform with a spatial cutoff window. Note that the scaling function and the wavelet function play different roles, where the low-frequency content of data is obtained by convolving the signal with the scaling function.
In the ablation section, will employ a specific discretization of the Meyer wavelet to generate our data. Additionally, we will explore the Haar wavelet method, which is implemented solely through spatial convolution kernels and scaling operations. The Haar wavelet, being the simplest form of wavelet, is particularly interesting because it uses piecewise constant functions to capture local features at varying scales, providing contrast to the smoother Meyer wavelet.
D.5 IMPLEMENTATION DETAILS
Syper-parameter In our experiments, the pretrained VAE from Stable Diffusion (Rom- bach et al. 2022). The VAE encoder has a downsampling factor 8 given an RGB pixel-based 4. All experiments operated in the fixed
image xx
E(x) has shape
latent space.
In Table S3 provide training hyperparameters for the image generation tasks on the two datasets.
For implementing training algorithm Algorithm [ the bump function is provided in eq. (19). For the classifier-free sampling the conditional generation task, the efg-scale is set to be 1.5.
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Supplementary Table S3: Hyper-parameters of DiT network.
CelebA 256
ImageNet
Model
DIT-I/2 (Peebles & Xie 2023
DIT-L/2 (Peblles Xie 2023
Ir
2e-4
le-4
Adam W optimizer (31 (32)
0.9, 0.999
0.9, 0.999
Batch size
32
240
# of epochs
500
900
# of GPUs
16
D.6 RESOURCE REQUIREMENT AND TIME COMPLEXITY
Resources Requirement. All experiments were conducted NVIDIA HI00 GPUs, with a total computational budget of approximately 3,000 GPU-hours.
Time Complexity. Our main focus is comparing the computational complexity of the K- amplitude flow with that of ordinary latent flows, we observe that during training, the additional computational overhead introduced by the K-aamlitude flow is minimal. From Algorithm I is evident that the only additional computational step is the discrete inverse K K-amplitude transform performed at each training iteration, while the remaining steps maintain the same complexity as the ordinary flow matching algorithm. For instance, when considering the Fourier transform, its com- putational complexity (( log N), where N denotes the length of the flattened image vector in the latent space.
For inference, from Algorithm 3 compared to ordinary latent flow, the only additional step to per- form the K amplitude flow is inverse K amplitude transform to set up the initial noise for gen- cration, and the remaining inference remains the same complexity. Thus, we expect a similar or slightly higher complexity than the ordinary latent flow during generation.
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E ALGORITHMS
In this section, we list three key algorithms.
Algorithm Training of K-Flow.
Require:
Scaling parameter k with maximum kmau. K-Flow transform F, inverse transform noise distribution p. target distribution
Normalize & to be in [0, 1]: k +- k/ k/kmar
Initialize parameters of uk
while not converged do
Sample scaling parameter k ~ ((,
Sample training example e ~ q. sample noise p
Calculate current flow position V according to K-Flow transform F,F and Equation (0
Calculate the conditional vector field V according to F, F- and Equation
Calculate the objective {(0) = |vx (vx; Vnllg following Equation 12)
optimizer.step((((0
end while
Algorithm Scaling Generation of K-Flow.
Require:
Scaling parameter k. K-amplitude transform F, inverse transform F-1, noise distribu- tion p in the K -m-apiiu space, target distribution Initialize pre-trained (0)
Sample one high-scaling noise Chigh ~ p. sample two independent low-scaling noise Clow ~ P.
Elow p
Vo = F (Elow, Ehigh
Po F- F-1(čo Chigh
for E [0. 1] do
4 ODEstep( Us 0), vo) V ODEstep( (vk 0), v
end for
return V1. L
Algorithm 3 Class-conditionss Generation of K-Flow with dropping.
Require: Pre-trained Un (6), conditioning class c, dropping time т, noise distribution p, guidance
parameter
1: Vo ~pp
p
2: for [0, do do
3: (x(-) (1 w) w) (-, wuulicc. {guided velocity) }
4: Vr +- ODEstepp Up (-), Mo)
5: end for
[r. do 7: V ODEstep(pa(...)) 1
6: for &
8: end for
9:
10: return V
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F MORE RESULTS
Discussion on the quantitative generation results We posit that standard generative modeling metrics, such FID and Recall, may not fully capture the specific advantages conferred by K-Flow, particularly concerning improvements in structural fidelity across the full spectrum of K-amplitudes. The benefits of our method become more apparent in downstream applications where high-fidelig detail is critical. For instance, as demonstrated in Appendix F.9, K-Flow exhibits stronger per- formance tasks like super-resolution and molecular assembly. This superior performance detail-dependent tasks supports the conclusion that K-Flow offers tangible advantages in generating high-fidelity structures, benefits that may not be fully reflected by conventional, parception-orientte metrics alone.
Comments on performance. Firsly, one of our primary motivations is indeed build upon the observation that some generative models, like DDPMs, appear to learn a process that progresses from low high frequencies. However, we note that this property is often an empirical post-hoc observation, tested with specific metrics, and its existence and characteristics within the flow match- ing paradigm are not well understood. From PCA perspective, a standard flow model fails to quantitatively exhibit the low-to-high frequency hierarchy. Thus, the contribution of our work is to transition this from an implicit, emergent property to an explicit and structured design principle. path, offering fine-
K-Flow provides generalized framework that formalizes this frequency-awar grained control that is absent standard models. Secondly, standard metrics like FID and Recall may not fully capture the benefits of improved structural fidelity across all K K-amplitudes.
F.1 MORE RESULTS ON UNCONDITIONAL GENERATION
We provide more results on the class-conditional generation using K-Flow in Figure S4
Supplementary Figure S4: Non-curated samples of our reversing scaling variant on ImageNet (cfg 1.5).
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F.2 CLASS-AWARD FID METRIC
We propose using the class-aware FID metric, defined as follows:
FID elssecocooonii = Ee)p(e) [FID(e)]
(25)
where for each class the FID is calculated by:
FID == (X, Xg) = Il - ugl2 Tr(2i, + 2n 2(222g)1/2)
(26)
Here, X7 and X denote the real and generated data subsets for class c, respectively. Based on FID(c), the Class-Dropping-Ratio (CDR) is defined by
(c) FiD
CDR Be)p(c)
where FIDber denotes the FID calculated for the flow model carried with the class condition for the whole process, and FIlD denotes the FID calculated for the flow model carried with the class condition for only subprocess (we keep the initial 30% of the inference time for the experiment). In practice, instead of computing the expectation over the entire class distribution p(c), we randomly select classes out of the total 1000 classes for evaluation.
30

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Published as a conference paper at ICLR 2026
F.3 ABLATION STUDIES ON CONTROLLABLE CLASS- GENERATION
In S we provide brief results the controllable class-conditional generation over ImageNet. Here, we would like to give more qualitative comparison between our model K-Flow and LFM.
K-Flow, Undrop
K-Flow, Drop
Supplementary Figure S5: Classifier-free guidance sampling of our Fourier-based K-Flow with hyperpa- rameter setting of cfg 3. In the right columns, the class condition is omitted for the last 50% of the scaling steps during inference, using the initial noise. be observed that the efg value increases and the duration of omitting the class condition decreases, the generated results appear nearly identical.
Supplementary Figure S6: Classifier-fres guidance sampling of our wavelet- based K-Flow with hyperpa- rameter setting of efg In the right columns, the class condition drooped for the last 70% of the scaling steps during inference, using the same initial noise. can observed that after dropping, K-Flow still pre- serves the high-scaling contents.
F.4 ABLATION STUDY ON WAVELET BASE
From Table S4 we tested two additional wavelet base, the discrete Haar basis (Haar 1911) and the discrete Myer basis (Meyer 1990) as a supplement the Daubechies wavelet (db6, Karam (2012) used in the main text. All three wavelets demonstrated comparable performance in terms of both the FID and Recall metrics.
Supplementary Table $4: CelebA-HQ 256.
Supplementary Table $5: 256.
Conditional ImageNet.
Model
FID
Recallt
CelehA-HQ 256
Model
FIDJ
Recall
K-Flow, Dey L/2
5.01
0.47
K-Flow, Wave (reverse)
23.06
0.58
K-Flow, Haar-DiT L/2
5.01
0.46
+ cfg=1.5
5.10
0.46
K-Flow, Db-DIT L/2 (three scales)
5.77
0.42
K-Flow, Wave-DiT L/2 (Ours)
17.8
0.56
+ cfg=1.5
4.49
0.44
K- K-Flow, Wave-DiT L/2 (three bands)
16.1
LFM, DVT L/2 (Dao et al, 2023
14.0
0.56
cfg=1.5
2.85
0.42
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Published as a conference paper at ICLR 2026
01
Supplementary Figure S7: Daubechies wavelet K K-amplitude with more components trained CelebA-2555
Data
Image
Noiso
Same Noise
Same Noise
Same Noise
Image Pair
Samo
Supplementary Figure S8: Pipeline of scaling generation (high scaling).
F.5 ABLATION STUDY ON SCALING PARTITIONS
Although the quality of face generation appears similar the naked eye, the model with three K- amplitude bands (the last row of Table S4) performed worse in terms of FID and Recall metrics. We provide the generated samples for qualitative evaluation in Figure SS7]
Reversing the K-amplitude Scaling. In Table [5) we also tested a counterintuitive scaling order: from high to low. This means generating high-frequency details first and then filling in the low- frequency components during the generation process. We find that the model can still produce images normally (Figure (54). with a better diversity (Recall) but lower quality (FID) compared to the low-to-high scaling approach.
F.6 IMAGE SCALING- LCALOLLCCN GENERATION
Preserving Low Scaling, Modifying High Scaling. We need highlight that in K-Flow, when modeling the flow from lower to higher scales, the noise at higher scales is used to predict the velocity the lower scale. This is determined by the nature of ODE flow. To this end, study by reversing the scaling direction in the Daubechies wavelet K-F K-Flow, and the pipeline is illustrated in Figure S8 In such reversed setup, we keep the low-scaling part the same noise while gradually denoising the high-scaling part.
conduct
The results are listed in Figure According to the six pairs of results, we can observe that the low-scaling part stays the same, like the background of the image and the gender and color of the people, while the high-resolution details facial expressions and outlook vary within each pair.
Remarks. Although the overall results are generally optimistic, some unexpected changes have been observed in the high-scaling parts. This may be attributed two factors:
1. The compressed latent space may mix high and low content present in the original pixel space.
2. The loss Equation vector field might not perfectly confined to the low-scaling part. The second factor might be mitigated by training flow (from high to low). we observe that fixing the low-scaling noise enables unsupervised editing of detailed high-scaling content.
12
may not be perfectly optimized, meaning that K-Flow localized
larger datasets. Furthermore, by training reversed K-Flow
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Supplementary Figure S9: Results of generation. We display six pairs of images, where each pair of images preserves the low scaling and differs in the high scaling.
ils
Supplementary Figure S10: I.FM editing by Algorithm
2
In Figure S9 we' ve tested the wavelet-bases K-Flow and observed similar results with the Fourier- based K-Flow.
This insight further supports our model's capacity to decompose the generative process into distinct frequency bands, where specific frequency bands can be independently controlled. This separation aids in achieving more detailed and deliberate modifications to generated data, adding a layer of precision and flexibility to the generative framework.
F.7 IMAGE RESTORATION
In this section, we evaluate the performance of the K-amplitude flow on image restoration tasks, including super-resolution and deblurring. These tasks typically involve reconstructing the high- frequency components of an image conditioned on the known low-frequency components. Unlike unsupervised editing based different scales, the performance of this experiment be quan- titatively measured using reconstruction metrics such as Peak Signal-to-Noise Ratio (PSNR) and Structural Similarity (SSIM).
Datasets and Baselines Our method benchmarked against standard diffusion and flow matching based restoration methods (see (Martin et al.) 2024) for detailed introduction). We evaluate all methods on CelebA dataset, with images resized to 128 x 128.
Algorithm. Our training-free free restoration method basically follows the efficient algorithm proposed in Martin et al. (2024) with two key changes adapted to K-amplitude:
1. The naive lincar interpolation step is replaced by our scaling interpolation formula Equa- tion (10).
2. Instead of starting restoration from pure noise, start t 0.5, since flow pri- marily denoises high-frequency components during the later period time. From the K- amplitude perspective, this strategic initialization point provides more informed starting state compared to conventional flow approaches while maintaining restoration quality.
We provide the algorithm details in Algorithm
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Published as a conference paper at ICLR 2026
Supplementary Table S6: Performance comparison on image restoration. this experiment, we pre-trained K flow with the same U-net architecture implemented (Martin et al. 2024).
Super-res.
Deblurring.
PSNR
SSIM
PSNR
SSIM
Degraded
10.17
0.182
27.67
0.740
PnP-Diff (Zuu et al. 2023)
31.20
0.893
32.49
0.911
PnP-GS (urrault et al.
2021
30.69
0.889
33.65
0.924
OT-ODE Pokle et al. 2023)
31.05
0,902
32.63
0.915
D-Flow [Ben-Hamu ci al.
2024
29.17
0.833
31.07
0.877
Flow-Priors ZZang
(2024)
28.35
0.717
31.40
0.856
PnP-Flow (Martin et al.
2024)
31.49
0.907
34.51
0.940
K-Flow (ours)
32.35
0.928
35.72
0.942
Clean
Degraded
Reconstructed (K-Flow, ours)
U
Supplementary Figure S11: Visualization for image restoration using K-Flow.
Algorithm PnP K-Flow.
Input: Pre-trained network o by K-Flow, time sequence (4n), either finite with tn n/N, N N or infinite with oo tn 1 and to 0.3, adaptive stepsizes (7a)..
Initialize: 2o E Rd.
for 0, 10 do
> Gradient step on the data-fidelity term in from 2, and noise through K- K-amplitude interpolation I0 En+11
2n = En 7n F(r,).
Dr,(
b- PnP step with restoration denoiser in (Martin
al. 2024)
return 2n+1
Results. We report benchmark results (following Martin (2024) for all methods on super- resolution and deblurring tasks, measuring average PSNR and SSIM on 100 test images, including super-resolution (with down sample rate x2). Results are averaged across 100 test images. From Table [S6] we see that the K-amplitude flow achieves pate-off-the-an (SOTA) quantitative results both the super-resolution and deblurring tasks. This superior performance without task-specific hyperparameter tuning can be attributed to our model's inherent frequency-awen design: both super- resolution and deblurring tasks primarily involve recovering values of the scaling parameter k ), which naturally aligns with the later stages of K-Flow's scaling- progressive generation process. From Figure SIT scaling components.
information (higher
we can clearly see how K-Flow restores the high
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Published as a conference paper at ICLR 2026
Supplementary Table S7: K-Flow against seven generative models on COD- 70D-Cluster1 with 5K, 10K, and all samples. The best results are marked in bold.
COD-ClusterI7-SK
COD-ClusterI7-Bn 17-10K
COD-Clusterl--AI
PM (atom).
PM (center).
PM (atom)
PM (center)
PM (atom)
PM (center)
GNN-MD
13.67 + 0.06
13.80 + 0.07
13.83 + 0.06
13.90 + 0.05
22.30
12.04
14,51 = 0.82
CrystalSDE- VE
15.52 + 1.48
16.46 0.99
17.25 + 2.46
17.86 + 1.11
17.28
0.73
18.92 0.03
CrystalSDE-
18.15 3.02
19.15
4.46
3.29
21.39
1.50
18.03
20.02
CrystalFlow-VE
14.87 7.07
13.08
4.51
16.41
+ 2.64
16.71
1 2.35
12.80
1.20
15.09 0.34
CrystalFlow-VP
15.71 2.69
17.10 1.89
19.39
= 4.37
16.01
3.13
13:50
0.44
13.28 0.48
CrystalFlow-LEnd
13.59 0.09
13.26 = 0.09
13.54
= 0.03
13.20 = 0.03
13.61 = 0.00
13.28 = 0.01
AssembleFlow
7.27 + 0.04
6.13 0.10
7.38
0.03
6.21 0.05
7.37 0.01
6.21 0.01
K-Flow (ours)
6.11 = 0.11
7.26 = 0.06
6.12 = 0.07
7.23 0.01
6.07 0.01
F.8 MOLECULAR ASSEMBLY
We consider another scientific task: molecular assembly. The goal is learn the trajectory on moving clusters of
molecular structures to the strongly-correlated structures.
Dataset and evaluation metrics. We evaluate our method using the crystallization dataset COD-Cluster17 (Liu et al. 2024b), a curated subset of the Crystallography Open Database (COD) Grazulis et al.) 2009) containing 133K crystals. We consider three versions of COD- Cluster17 with 5K, TOK, and the full dataset. To assess the quality of the generated molecular assemblies, we employ Packing Matching (hishomi
Motherwell] 2005), which quantifies how well the generated structures align with reference crystals in terms of spatial arrangement and packing density. Following (Liu et al. 2024b), we compute PM at both the atomic level (PM-atom) and the mass-center level (PM-center) (Chisholm & Motherwell 20055).
Baselines. We evaluate our approach against GNN-MD (Liu et al. 2024b), variations of Crys- taISDE and CrystalFlow (Liu et al. 2024b). and the state-of otate-of-the-arr AssembleFlow (Guo et al. 2025). CrystalSDE--EE/
model diffusion via stochastic differential equations, while CrystalFlow- VE/VP use flow matching, with VP focusing variance preservation. CrystalFlow-LERk employs
linear interpolation for efficiency. AssembleFlow (Guo et al. 2025) enhances rigidity modeling using an inertial frame.
Main results. The main results in Table show that K-Flow outperforms all baselines across three datasets. Building on AssembleFlow's 's rigidity modeling. K-Flow decomposes molecular pairwise distances via spectral methods and projects geometric information from RR3 and SO accordingly. This approach achieves consistently superior packing matching performance.
35


Reviews:
[{"review_id": "0hZRksStxM", "reviewer": "Reviewer_PPuw", "summary": "Disclaimer: I found the paper hard to understand, I did my best to summarize what I understood.\n\nThis paper propose to perform a change of basis (Fourier, Wavelet, PCA) on the data, cluster the new representation into K ordered sub-basis and to do diffusion for each sub-basis in sequence (in the Fourier case we can think of it as frequency bands).\nTo clarify, what is typically referred as time $t$ in diffusion is now a range $t'\\in[0,K]$ where $k=\\lfloor t'\\rfloor$ and $t=t'-k$.\nIn this new setting the new definition of time interleaves the classical time definition with ordered sub-basis.\n\nThe claimed benefit of such a proposal is to allow for more steerable generation, where the steerability is determined by the choice of change of basis and the sub-basis on which to do diffusion.\nThe evaluation is performed on generative tasks and an ablation demonstrates steerability by sampling only in some of the sub-basis or removing the label during inferences.", "questions": "1. The gains obtained from K-Flow appear marginal at best. Is it because the classical diffusion noise is already implicitly doing a K-Flow and therefore no gains are achieved from expliciting doing a K-Flow?\n2. On the steerability front, it would be interesting to see differences k-band steering between Fourier, Wavelet and PCA. I assume steerability is heavily influenced by the choice of basis. Did you do such experiments?\n3. Is the neural network actually conditioned on $(k,t)$ as a tuple or $t'$, or are there $k$ different networks?", "limitations": "", "rating": 2, "confidence": 3, "soundness": 3, "presentation": 1, "contribution": 3, "strengths": "1. The idea seems novel and interesting.", "weaknesses": "1. The presentation/writing of the paper is very hard to follow, I had to read 3 times to begin understanding it (or so I assume). In particular, it's overloaded with notation complexities that could probably be simplified to make it more accessible.\n2. The results are not particularly striking for unconditional and conditional generation, matching existing SotA but not apparently showing clear gains.\n3. The choice of CelebA is questionable for unconditional generation as it is quite unimodal in the first place, unlike ImageNet.\n4. The resolution used for ImageNet is not mentioned in the main paper, but I found near the end of the appendix that it must be 256x256.\n5. The PCA variant of K-Flow is missing from Table 2.\n6. The paper lacks a discussion of why the method does not perform better on generation tasks. \n7. In terms of steerability, the results do not seem particularly useful. In one case the class label is dropped when $k$ a certain value (30%) and is still able to generate good samples (but since the label is always available I fail to see the relevance of this experiment). In another case, some sub-basis are frozen while some other basis are sampled from resulting in changing either low-frequency or high-frequency details. It works but I don't understand under what circumstances one would want to do this. It gives the feeling to be a solution to a non-problem or maybe better examples of steerability are needed to inspire readers.\n8. Line 333, you mention \"Date-dependent PCA\", do you mean \"Data-dependent\"?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_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": "DnRQZTNi7E", "reviewer": "Reviewer_m9JR", "summary": "This paper introduces a generative model based on flow matching that works in the K-amplitude space. The model leverages three K-amplitude transformations, Fourier, Wavelet, and PCA, to construct its generative process. Experimental results show that the proposed method achieves performance comparable to the baselines on both unconditional and conditional image generation on natural images and the molecular assembly dataset. Furthermore, the authors show that their approach enables multi-scale control over the generated information.", "questions": "Please consider clarifying the following point:\nShould there also be a summation in line 128, similar to the one in line 123?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The main idea sounds novel and is supported by compelling experiments. The paper is well-written and easy to understand.", "weaknesses": "As discussed in your Related Works section, the following papers share significant similarities with your approach. I recommend including them as baselines for experimental comparison:\n\n[1] Tian, Keyu, et al. “Visual Autoregressive Modeling: Scalable Image Generation via Next-Scale Prediction.” Best Paper, NeurIPS 2024.\n\n[2] Ren, Sucheng, et al. “FlowAR: Scale-Wise Autoregressive Image Generation Meets Flow Matching.” ICML 2025.\n\nIncluding these two baselines in your comparison would substantially improve your empirical evaluation. I would be happy to potentially raise my score.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_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": "1j3UmvgGXi", "reviewer": "Reviewer_V9Lm", "summary": "This paper proposes a novel generative modeling paradigm called **K-Flow**.  The core idea is to replace the traditional **time variable** \\( t \\) in *Flow Matching* with a **scale parameter** \\( k \\).  This enables the model to evolve from noise to data **along a multi-scale trajectory** (e.g., from low to high frequencies).  The authors instantiate K-Flow with three types of **K-amplitude transformations**: **Fourier Transform**, **Wavelet Transform**, and **PCA**.  They claim that this approach provides a **more natural multi-scale modeling framework** and allows for **finer-grained controllable generation**.", "questions": "1. **Regarding CDR:** The CDR formula in Appendix F.3 ($FID_{bef} / FID_{aft}$) seems to contradict the conclusions in the main text and the visual evidence in Figure 4. According to this formula, LFM's CDR = 3.25 would imply that its $FID_{Drop}$ quality is much better than $FID_{Undrop}$. Could this formula be a typo (e.g., with the numerator and denominator swapped)?\n\n2. **Regarding performance:** Why does the K-Flow (Wavelet) variant perform the best in Table 1 but fall far behind the LFM baseline in Table 2? Does this suggest that wavelet transforms may not be suitable for complex class-conditional tasks like ImageNet?\n\n3. **Regarding hyperparameters:** Why does increasing the number of K-amplitude bands (from 2 to 3) improve performance on the LSUN dataset (App F.2) but decrease performance on the CelebA-HQ dataset (App F.6)? Does this indicate that K-Flow is highly sensitive to this hyperparameter and that its optimal setting depends on the dataset?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 1, "presentation": 2, "contribution": 2, "strengths": "1. **Novel framework:** Replacing the “time” variable \\( t \\) in flow models with a “scale parameter” \\( k \\) is a novel and conceptually elegant idea.  \n2. **Generality:** The framework is general and can be combined with various linear transformations such as Fourier, Wavelet, and PCA.  \n3. **(Limited) Qualitative results:** Although the CDR metric has certain flaws, the visual robustness of K-Flow under the “Drop” condition in **Figure 4**, as well as the controllable generation results in **Figure 5**, are qualitatively interesting.", "weaknesses": "1. **Logical contradiction in the CDR metric:**  \n   In Sec. 4.2, the CDR metric is used to demonstrate controllability, but its mathematical definition in Appendix F.3 directly contradicts both the textual explanation in the main paper (“higher CDR indicates performance degradation”) and the visual evidence in **Figure 4**.  \n   This inconsistency invalidates the quantitative conclusions of the ablation study. The authors must correct the metric (most likely the formula is inverted) and reanalyze the results.\n\n2. **Unexplained performance inconsistency:**  \n   The paper must address why the **K-Flow (Wavelet)** variant performs best on **Table 1 (CelebA-HQ)** but performs poorly on **Table 2 (ImageNet)**—significantly worse than the LFM baseline.  \n   This raises serious concerns about the method’s robustness and generalization capability.\n\n3. **Hyperparameter fragility:**  \n   The authors must explain why the number of K-amplitude frequency bands (e.g., 2 vs. 3) leads to completely opposite performance trends across datasets (Appendix F.2 vs. F.6).  \n   This indicates that the method is fragile and sensitive to hyperparameter tuning, undermining the claim of “natural” multi-scale modeling.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_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": "CpsWcloKGQ", "reviewer": "Reviewer_ZFwo", "summary": "This paper proposes K-Flow, a novel generative modeling paradigm that conducts flow matching in the K-amplitude domain—where the scaling parameter k organizes projected coefficients (e.g., frequency bands) and \"amplitude\" refers to the norm of these coefficients. By instantiating K-Flow with three classic transformations (Fourier, Wavelet, PCA), the work addresses a key limitation of conventional flow matching (FM) models: the lack of explicit utilization of natural data’s frequency hierarchy. Through extensive experiments on unconditional/conditional image generation, controllable generation, and image restoration, the paper demonstrates K-Flow’s competitive performance and unique steerability. The work’s theoretical grounding (six core properties of K-Flow), architectural flexibility (backbone-agnostic design), and practical value (training-free restoration, unsupervised frequency editing) make it a meaningful contribution to generative modeling research, well-aligned with ICLR’s focus on innovative and rigorous methods.2. Strengths", "questions": "Regarding the observation that MLP heads outperform Transformer heads in text adherence, could you provide a more in-depth analysis? For example, from the perspective of token-level alignment between text prompts and generated images, or exploring whether the attention mechanism in Transformer heads interferes with text-image alignment?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "Innovative Theoretical Framework with Clear Motivations:K-Flow fills a critical gap in existing FM literature by leveraging the inherent frequency structure of natural data. Unlike conventional FM models—whose frequency progression is unstructured and unquantified—K-Flow formalizes the K-amplitude space, where k systematically organizes frequency bands and amplitude reflects signal energy. The six properties of K-Flow (from theoretical foundations like scale-energy alignment to practical benefits like explicit steerability) provide a rigorous framework for understanding and extending the paradigm, distinguishing it from ad-hoc frequency-aware modifications.", "weaknesses": "Lack of Direct Comparison with Frequency-Aware Diffusion Models\nWhile K-Flow is positioned as a flow-matching innovation, the paper does not explicitly compare it to frequency-aware diffusion models. Adding such comparisons would clarify K-Flow’s advantages in terms of both performance and computational efficiency, especially in scenarios where diffusion and flow-matching paradigms overlap.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_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": {}}]

Author rebuttal:

Decision: Accept (Poster)