CacheFlow: Fast Human Motion Prediction by Cached Normalizing Flow
Authors: Takahiro Maeda
Jinkun Cao
Norimichi Ukita
Kris Kitani
Venue: Accepted by TMLR ()

Abstract: Many density estimation techniques for 3D human motion prediction require a significant amount of inference time, often exceeding the duration of the predicted time horizon. To address the need for faster density estimation for 3D human motion prediction, we introduce a novel flow-based method for human motion prediction called CacheFlow. Unlike previous conditional generative models that suffer from poor time efficiency, CacheFlow takes advantage of an unconditional flow-based generative model that transforms a Gaussian mixture into the density of future motions. The results of the computation of the flow-based generative model can be precomputed and cached. Then, for conditional prediction, we seek a mapping from historical trajectories to samples in the Gaussian mixture. This mapping can be done by a much more lightweight model, thus saving significant computation overhead compared to a typical conditional flow model. In such a two-stage fashion and by caching results from the slow flow model computation, we build our CacheFlow without loss of prediction accuracy and model expressiveness. This inference process is completed in approximately one millisecond, making it 4$\times$ faster than previous VAE methods and 30$\times$ faster than previous diffusion-based methods on standard benchmarks such as Human3.6M and AMASS datasets. Furthermore, our method demonstrates improved density estimation accuracy and comparable prediction accuracy to a SOTA method on Human3.6M. Our code and models are available at \url{https://github.com/meaten/CacheFlow}.

Published in Transactions on Machine Learning Research (01/2026)
CacheFlow: Fast Human Motion Prediction by Cached Nor- malizing Flow
Takahiro Maeda
takahāro maeda.k-) 1.9mmlllll.mmmmmmmmmmmmmmmmmmmm toshiba jinkunc Gandrew. cmvu. edu ukita@toyotts ti. ac.jp kmkittaii Bandrew. cmau. edu
Jinkun Cao²
Norimichi Ukita'
Kris Kitani
Toyola Technological Institade
Robolics Institute, Carnegie Mellon Universily
arX arXivisubmit/7208555 [cs.CV] 29 29Jan Jan 2026
Reviewed on OpenReview: https: openreview. net/ forum? id= y8dGdBJkka
Abstract
Many density estimation techniques for 3D human motion prediction require a significant amount of inference time, often exceeding the duration of the predicted time horizon. To address the need for faster densily estimation for 3D human motion prediction, we introduce novel flow-based method for human motion prediction called CacheFlow. Unlike previous conditional generative models that suffer from poor time efficiency, CacheFlow takes advantage of an unconditional flow-based generative model that transforms Gaussian mixture into the density of future motions. The results of the computation of the flow-based generative model can be precomputed and cached. Then, for conditional prediction, we seek a mapping from historical trajectorics to samples in the Gaussian mixture. This mapping can be done by much to typical conditional flow model. In such two-stage fashion and by caching results from the slow flow model computation, we build our CacheF without loss of prediction accuracy and model expressiveness. This inference process is completed in approximately one millisecond, making it 4x faster than previous VAE methods and 30x faster than previous diffusion-based methods on standard benchmarks such as Human3.66 6M and AMASS datasets. Furthermore, our method remonstrates improved density estimation accuracy and comparable prediction accuracy a SOTA method on Human3.66 Our code and models available at https://github. htt.aaa..commmattt////cceeel....
light model, thus saving significant computation overhead compared
1
Introduction
The task of 3D human motion prediction is forecast the futurr 3D pose sequence given an observed past sequence. Traditional motion prediction methods are often based deterministic models and can struggle to capture the inherent uncertainty in human movement. Recently, stochastic approaches have addressed this limitation. Stochastic approaches allow models to sample multiple possible future motions. Stochastic human motion prediction methods utilize conditional generative models such as generative adversarial networks (GANs) (Goodfellow et al. 2020), variational autocncoders (VAEs) (Kingma & Welling 2014). and denoising diffusion probabilistic model (Ho et al. 2020). However, many stochastic approaches canmot explicitly model the probability density distribution.
Conversely, density estimato-base approaches explicitly model the probability density distribution. In safety- critical applications such aulonomous driving (Paden et al. 2016) and human-robol interaction (Koppula)
Bütepage et al.] 2018 density estimate can represent all possible future motions (not just a few samples) by tracking the volume of density. It can be used to derive guarantees
& Saxena 2015 Lasota & Shah
2017
on safety (Mainprice & Berenson
2013
Sanderud et al. 2015
Tisnikar et al. 2024
*Currently Tushiba Corporation.

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Published in Transactions on Machine Learning Research (01/2026)
Previous Methods (-100 inference)
CacheFlow (1.3 ms inference)
Generation slive Model
Flow- Flow-baser based Model
Conditional
Precomputed ted
Observation
Fixed noise
Predictions
Obscrvation Conditional base density
Predictions
Sampled noise
Procomputed and eached latents
Probubility distritution
Figure Previous methods vs. Our CacheFlow. Previous methods of stochastic motion prediction generate multiple future motions by sampling noises from the fixed source in an ad hoc manner. In contrast, CacheFlow uses the precomputed and cached moeenn pairs from an unconditional flow-based generative model. Thus, the computation of the unconditional flow be skipped inference. One achieve fast inference by selecting predictions from these cached pairs.
However, previous density estimation suffers from high computational cost. The expensive computational cost. can prohibit applications real-time use-cases, especially with high dimensional data such as human motions. For instance, kernel density estimation (KDE) (Rosenblatt) 1956 Parzen 1962) requires an exponentially growing number of samples for accurate isatiin Concretely, more than one trillion samples are required for accurate KDE over 48-dim pose over 100 frames of human motion prediction (Silverman 1986).
In contrast to traditional KDE, recent parametric density estimation approaches use conditional Row-based generative models, including normalizing flows (Rezende & Mohamed] 2015 Tabak & Turner 2013 Tabak & Vanden-Fijnden, 2010) and continuous normalizing flows (Chen et al. 2018). These flow-based generative models ("flwwbased model" for brevity) directly estimate the density to avoid Lime-consuming sampling required in KDE. However, inferring the exact probability of possible future motions remains computationally expensive. This is because capturing the full shape of the distribution requires evaluating the probabilities of many potential future motions.
To address this computational limitation, we propose a fast density estimation method based on a flow-based model called 'CacheFlow". Our CacheFlow utilizes an unconditional flow-based model for prediction, as illustrated in Fig. [ Since the unconditional flow-based model is independent of past observed motions, its calculation be precomputed and skipped at inference. This precomputation omits large portion of computational cost. To achieve further acceleration, transformation between a lightweight conditional base density and the density of future motions. At inference, the density of future motion is estimated by computing the weight conditional base density and combining it with the precomputed results of the flow-based model. The inforence of our method is approximately one millisecond.
unconditional flow-based model represents
CacheFlow demonstrates comparable accuracy previous methods on standard stochastic human motion prediction benchmarks, Human3.6N. 6. (Ionescu et al. 2013) and AMASS (Mahmood et al. 2019). Furthermore, our method estimates densily more accurately than previous stochastic human motion prediction methods with KDE. CacheFlow shows improved computational efficiency, making it well-suited for real-time applications. The contributions of this paper are fourfold as follows:
We introduce a novel fast density estimation called CacheFlow on human motion prediction.
2. We can sample diverse future motion trajectories with explicit density estimation, and experimen- tally confirm that our method can estimate accurate density.
Our method achieves comparable prediction accuracy to other computationally intense methods on several benchmarks.
2

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Published in Transactions on Machine Learning Research (01/2026)
2
Related Work
2.1 Human Motion Prediction
Detorministic approaches. Early approaches on human motion prodiction (Aksan et al.
2021 Bouazizi 2016 Li al.
et al.
2022 Fragkiadaki al., 2015; Butepage al.,
2017 Guo et al. 2023 Jain et al.
2018
focused on deterministic settings. They predict the most likely motion sequence based
the past
motion. A wide range of architectures were proposed including multi-layer perceptron (Guo et al.
2023
recurrent neural networks (Fragkiadaki et al. 2015 Jain et al. 2016 Martinez et al., 2017a;
Gui Gui et al.
2018
Pavllo et al.
2018: Liu et al. 2019)), convolutional neural networks (Li et al. [2018] Medjaouri & Desai 2022).
transformers (Aksan ct al. 2021 Cai ct al. 2020; Gartinez-González ct al.] 2021). and graph neural networks
(GNNs) (Mao et al. [019 Li et al. 2020 Dang el al. 2021 Li et al. 2021] J. GNN can account for the explicit
tree expression of the human skeleton, while other architectures implicitly learn the dependencies between
joints.
Stochastic approaches. To capture the inherent uncertainty in human movementss recent works have focused on stochastic human motion prediction to predict multiple likely future motions. The main stream of stochastic methods use generative models for the purpose, such as generative adversarial networks (GANs) (Barsoum
ct al. 2018 Kundu et al.
2019), variational autoencoder (VAE) (Walker et al., 2017 Yan et al. 2018
Mao al. 2021 Cai et. al.
2021 and denoising diffusion probabilistic model (DDPM) (Barquero et al,
2023
Chen et al. 2023; Wei et al.
2023 Sun & Chowdhary 2024). . To improve the diversity of predictions,
diversity-promoting loss (Mao et al.
2021 Barquero et al. 2023) or explicit sampling techniques (Wei et al.
2023) were proposed. In contrast to generative models, anchrr-based methods (Xu et al., 2022 2024 learn fixed number of anchors corresponding to each prediction to ensure diversity. However, most stochastic methods cannot describe the density of future motions explicitly. This prevents exhaustive or maximum likelihood sampling for practical applications. On the contrary, our method allows for explicit density estimation using normalizing flows (Kobyzev et al. 2020
2.2 Density Estimation
Density estimation asks for explicit calculation of the probability for samples from distribution. Density estimation is derived by non-parametric or parametric methods.
Non-paracctric Approach. The representative non-maraectric density estimation is kernel density estimation (KDE) [Rosenblatt, 1956 Parzen 1962). KDE can estimate density by using samples from generative models. However, KDE requires a large number of samples for accurate estimation. Therefore, it often cannot run in real-time.
Parametric Approach. As representative parametric model, Gaussian mixture models (GMMs) parametrize density with several Gaussian distributions and their mixture weights. Its nature of mix- ing Gaussian priors limits its ability to gemeralize to complex data distribution. Another parametric approach with more expressivity is normalizing flows (NFs) (Rezende Mohamed] 2015
generative models (Kobyzev et al.
2020). By a learned bijective process, 2013
Tabak & Turner
Tabak & Vanden-Fijnden, 2010)
transform a simple density like the standard normal
distribution into
complex data density. Recently, con-
timuous normalizing flows (CNFs) (Chen et al. 2018
Grathwohl et al.
2019 achieve more expressive density
than standard normalizing flows via an ODE- based bijective process. While training of CNFs is inefficient due to the optimization of ODE solutions, an efficient training strategy named flow matching (Lipman et al., 2023) was proposed. FlowChain (Maeda & Ukita 2023) was proposed for fast. and efficient density estimation in human trajectory forecasting. FlowChain improves the inference time efficiency by reusing results from the conditional flow-based method while the past sequences are similar. However, with significantly different past. sequences, Chownssin's efficiency can't hold anymore. Unlike Flow our method can perform fast and efficient inference regardless of past sequences.
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2.3 Diffusion- Motion Modeling
Recently, diffusion-based generative models have achieved state-of-the-arl performance in synthesixing diverse and high-fidelity human motion. The Motion Diffusion Model (MDM) (Tevet et al. 2023) demonstrated that diffusion processes can model complex kinematic distinctness, capable of tasks ranging from text-to-motion to motion completion. Concurrently, approaches inspired by Large Language Models, such as MotionGPT (Jiang et al. 2023) and MotionIM (Seff al. 2023). treat motion sequences as discrete tokens, leveraging the transformer architecture capture long-teryy semantic dependencies.
While these methods excel in generation quality and semantic control, they often incur high computational costs during inference. Diffusion models typically require multi-step iterative denoising, and LLM--aased approaches rely on heavy autoregressive sampling, In contrast to these heavy generative frameworks, our work focuses specifically on the lftciency-latency trade-off. While we share the goal modeling complex motion distributions, Cac leverages the tractable likelihoods of Normalizing Flows and introduces caching mochanisms to ensure rapid inference suitable for real-time prediction scenarios, avoiding the computational bottleneck of iterative difTusion or large-scale transformers.
3 Preliminary
3.1
Problem Formulation
The task of human motion prediction aims to use a short sequence of observed humnn motion to predict the future unobserved motion sequence of that person. Human motion is represented by a sequence human poses in a pre-defined skeleton format of 3D locations of J joints, X Jx3, As input to our model, we have the past (history of) human motion as sequence c == X2, X4] E "xxx3x3 over H timesteps. To predict the future human motion sequence of F timesteps, we can formulate the problem using the conditional probability function, p(X(c), where X [X#+1; X [H.,] /xx33 Similar to the stochastic human motion prediction paradigm, the method should also allow for sampling multiple future sequences {X1,... .X.,,..,,, X,) from p(X lc). The focus of our work is to accelerate the inference time of estimate and sampling of the conditional density function ((.c).
of conditional generation Jx3,
3.2 Normalizing Flow
Normalizing flow (Rezende & Mohamed] 2015
Tabak Turner 2013 Tabak & Vanden Vanden-Eijnden 2010) is a
generative model with explicit density estimation. It follows a bijective mapping fo with learnable parameters 0. It transforms simple base density (=) such as Gaussian distribution into the complex data density p(æ). We can analytically estimate the exact probability via the change-ofvvariables formula follows:
= fo(z),
2 = (z).
(1)
p(x) = (etz))detJJn (z)1-
(2)
where (z) afe is the Jacobian of fe at z. The parameters of fa can be Icarned by maximizing the likelihood (or conditional likelihood) of samples & from datasets or minimizing the negative log-likelihood LNLI log When and are latent. codes, normalizing flow is transformed into latent normalizing flow. We follow this pattern in our method. We encode the past human motion into by encoder network E and decode it by decoder network D:
az
x E(X) = D(x). x ~ Rdd Fx333
(3)
The encoder and decoder are trained by reconstruction. In the later part of this paper, for simplicity, we discuss the method at the latent representation level and model the conditional generation task as p(x)cc..
3.3 Continuous Normalizing Flow (CNF)
Continuous normalizing flow (CNF) CChn et al. 2018 Grathwohl et al. 2019 normalizing flow variant based on ordinary differential equation (ODE). CNF defines i-continuous path 2, between the base density
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Unconditional Base Density
Unconditional Estimated Density
Conditional base density
Conditional Estimated Density
(2)
pxx)
qg (2|c)
p(xx
Unconditional Prediction Tate)
Conditional Prediction 9g(2()) Jya
mmm
(a) Precomputation
(b) Inference
Zk Ir (Zx)
X
X K
Store triplets
Figure 2: Overview of our CacheFlow. Our method utilizes the unconditional flow-based model fe. This fa maps the lightweight conditional base density (=(=]c into future motion density p(r]c). In this formulation, the flow- based model is independent of past motions. Thus, we can precompute the unconditional flow-based model. These results are cached as K triplets as shown in (a). Due to the precomputation, we can skip the inference of fe and omit a large portion of the entire computation. At inference, density estimation is achieved by only evaluating the lightweight conditional base density qa (zx(cc and combining it with the stored K triplets as shown in (b).
space Zo q(2) and the data space 21 p(æ)). This 2, defined by the parameterized vector field da. =z()) The data 21 is generated via numerical integration of vector field vo(2)) follows:
x = 21 = 20 + f vo vo(z))dl.
(4)
The CNF transformation Eq. (4) is denoted as x = fe( for brevity. Although CNF can trained by
minimizing negative ligliiollhood, it is time-consuming due to the numerical integration of ODE.
3.4 Flow Matching
In order to train the parameterized vector field efficiently, one can leverage the flow matching (Lipman et al, 2023) strategy. As a new training strategy, Flow Matching avoids the numerical integration of ODE by directly optimizing the vector field ug (z).. The objective of flow matching is match the parameterized vector field 02(2,) to the ground truth vector field u(2,) via mean squared error follows:
LFM 1. (lvo(z)) u(2))11²,
(5)
where T(0,1) 1) is distribution ranging from to 1, such uniform distribution.
However, cannot obtain the ground truth vector field u(z;) directly. Lipman et al. (2023) suggest defining
the conditional ground truth u(z,(²) 21) instead. Specifically, it is modeled as straight vector field 2 in
Rectified Flow [Liu et al. 2023). This is trained by the following objective:
LCFM = E...T..))......lll (1va - u u<=r12))11
(6)
The gradients of LFM of Eq. 5 and LCFM of Eq. 6 are identical w.r. 0. We exploit expressive CNFs with efficient Flow Matching training to estimate the future motion density p)z
4 Proposed Method
4.1 Overview of CacheFlow
As mentioned Sec. 3.2, we propose a flow-based method on a latent representation using frozen & and D. We estimate the future motion density p(x/cc by transforming a conditional base distribution (s (2)2c),, This q is conditioned on the past motion c. Then we can sample predictions æ ~ p(ælc) for stochastic human motion prediction. Most traditional approaches based conditional generative models trivial source
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Published in Transactions on Machine Learning Research (01/2026)
distribution, often simple Gaussian. However, redefine the source distribution to be more informative and directly regressed from past motions. This allows us to develop much lighter and faster model for predicting future movements.
To build this informative conditional base distribution gø, we would incorporate an unconditional flow-based model fo æ = fo(z) that maps latent variable into motion representation æ. To understand how te and fo are connected, we first reparametrize the future motion density p(ærle) by change of variables of probability equation follows:
Oz dx (f((z)
p(alcc = (((ec)
det
= <(|c)) det
(8)
Oz
det
(9)
This parametrization trick differs from the widely-used conditional density formulation (Winkler et al. where e is conditioned to the flow-based model fe. In this formulation, only the conditional base density g(= c) varies depending on c during inference, whereas the unconditional flow-based model x = f=()) and the Jacobian fdet.f (z)l are kept same during inference and thus can be reused as-is once calculated.
2019
Therefore, we could precompute the mapping results and Jacobians of an unconditional flow-based model fo- We cache the triplets t {z, (2)1= for later reuse in the inference stage, as shown in Fig. 2
Then, during inference, we design a new trick to reuse the cached triplets by associating them with the specific conditions of the past motion sequences, as shown in Fig. bb). Now, instead of a typical conditional generative model, e.g,, conditional normalizing flow, we only need a light weight, model to model the conditional base density (o(2(c) and achieve similar expressivity. We could finally estimate the future motion density by p( c) = qo (z(c) d (z)1-1 The method is summarized as pscudocode in Algorithm In the following paragraphs, we elaborate on the details of our method.
4.2 Precompute Unconditional Flow-based Model
As the first step of our method, we use the human motion dataset to learn an unconditional flow-based model fo- From this unconditional human motion prediction model, we will collect the triplets t {zz, Jdet (=)1- a} for later use. This part is illustrated in Fig. 2 a).
In our implementation, we built the unconditional flow model by CNFs due to its proven expressivity for predicting human motion. The unconditional model is trained to predict a fixed-length future motion æ given a noise sample z sampled from a source distribution q(z), a standard Gaussian distribution:
So Rdd RRd
(10)
Because is sampled from a known distribution and normalizing-for models are deterministic with reversible bijective transformation, normalizing flow with the flow matching objective described in Eq. 6 Then we collect K samples denoted by the triplet & = {* |det (=x))_",, }. Triplets are collected by applying the inverse transform of fo to
could know the density of cach {z, x)) pair. We train the unconditional continuous
ground truth future motions in the training split. These triplets are cached for fast inference as described
in Sec. 4.3 This caching operation is different from anchor- based methods (Xu et al. 2022
2024 since
Cache Flow caches all motions of the training split.
4.3 Conditional Inference by CacheFlow
In previous methods, conditional human motion prediction typically requires a conditional generative model. For instance, is a conditional flow-based or diffusion model. These models usually have poor time efficiency due to delicate but heavy architecture. Instead, inspired by Eq. (g), we can reuse the results of unconditional inverse transformation triplets l {zx, Jdet. (2x) -1 m. ). Thus, we perform conditional inference
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Algorithm 1: Precomputation and Inference of CacheFlow.
Input: Past motion
Output: Estimated density p(zz(c)
// Precomputation. This does not count for inference time.
for each future motion X5 in the training datasel do
2& f EXxx)
+ (xx)
Calculate (26))-1
Store triplet {zs, Jdet 1o xx
end
// Fast Inference
for each triplet { to (zx)) [x, do
(o(2x|c) t- Em=1 Wm (m()) (c)) p(xx(c) [c) f- 2o (el (zx
end
by only evaluating a conditional base distribution g(z(c). We model this conditional base distribution by a learnable model, thus we denote it as (z(c). This model can be very lightweight since the unconditional transformation fo gives enough expressivity. (o(2(c) runs much faster than a typical conditional generative model for human motion prediction. This part is illustrated Fig. bb).
In our implementation, (z(z)c) is constructed as a parametrized Gaussian mixture (N((m (f)) with M mixture weights Wm (c), such that Em=1 1mm = 1. Each Pm and om are regressed based on the past motion c. We use a lightweight single-layer RNN for regression to determine the GMM composition. This RNN takes the past motion directly as input, and regress the Wm: o Therefore, & mcans the parameter of this RNN in our method. Although the unconditional flow-based model fo and the conditional base density 9 be trained separately, found that jointly training fo and 9 improves model performance. We train the joint model by summation of lig-liiel for qs and flow matching for fo explained in Eq. 6 follows:
C = - lo + LCFM-
(11)
With joint learning, fo learns an easy mapping for the conditional Gaussian mixture qs-
With d constructed, during inference, we can estimate the conditional density p(x|c) by connecting with precomputed triplets le {Zk, Jd (zx) -1, mx,
p(z)/c) 90 (etttc)deeeTtrr.....cc (zx))
(12)
By this inference process, we could optionally generate a future human motion sequence æ by retrieving a sample z from Is with the past motion sequence as the condition. However, de describes a continuous distribution and the stored triplets cannot cover all samples. Therefore, in practice, predicted motion zg- is selected by the nearest neighbor of the sampling outcome go to the stored triplets:
k² argming 1| - =|1. s.t. (4x = {*x, (d m.))) {o2ecc)
(13)
where A= is the selected index of the triplets for prediction. By this design, we can sample an arbitrary number of likely future motion sequences by selecting the neighbors of samples z (z(c)).
5 Experimental Evaluation
Datasets. We evaluate our CacheFlow on Human3.60 [Inescuu al. 2013 and AMASS Mahmoood al., 2019). Human3.6N 3M contains 3.6 million frames of human motion sequences. Human motions of 11 subjects
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Table 1: Quantitative comparisons over the stochastic human motion prediction metrics on Human3.00 and AMASS datasets. Lower is better for all metrics except APD. The reported inference time is when method
finishes generating 50 prediction samples from receiving the past motion.
Human3.6M [Tonescu et: al. 2013)
AMASS
MMahmood et al. 2019)
Inference Time(ms]]
APD1
ADE4
FDE4
MMADE. MMFDEL
APDI
ADEL
FDE4 MMADE4 MMFDRL
HP-GAN Barsouu et al.
2018
7.214
0.858
0.867
0.847
0.858
DSF
Yuan & Kilani 2019)
9.330
0.493
0.592
0.550
0.599
DeLiCAN Gurmmurt Bury by of al.
2017
6.509
0.483
0.534
0.520
0.545
GMVAE
Dilokthankkul et al.
2016
6.769
0.461
0.555
0.524
0.566
TPK
Wa ker et al.
2017
6.723
0.461
0.560
0.522
0.569
9.283
0.656
0.675
0.658
0.674
30.3
MT-VAE
Yan et
2018)
0.403
0.457
0.595
0.716
0.883
BoM Bhallacharyya el.
2018
6.265
0.448
0.533
0.514
0.544
DLow
Yuan & Kitani
2020
11.741
0.425
0.518
0.495
0.531
13.170
0.590
0.612
0.618
0.617
30.8
MultiO
Ma al] 2022
14.240
0.414
0.516
GSPS Mao et al. 2121
14.757
0.389
0.496
0.476
0.525
12.465
0.563
0.613
0.609
0.633
5.1
Motron
Salzmann ct al.
(022)
7.168
0.375
0.488
0.509
0,539
DivSamp Dang ct | (022)
15.310
0.370
0.485
0.475
0,516
24.724
0.564
0.647
0.623
0.667
5.2
BeLFusion Barquero et al.
2023
7.602
0.372
0.474
0.473
0.507
9.376
0.513
0.560
0.569
0.585
449.3
BeLFusion-D
5.777
0.367
0.472
0.469
0.506
7.458
0.508
0.567
0.564
0.591
39.3
HumanMAC (Chen et al.
2023
6.301
0.369
0.480
0.509
0.545
9.321
0.511
0.554
0.593
0.591
1172.9
CoMusion (Sun & Chowdhary 2024)
7.632
0.350
0.458
0.494
0.506
10.848
0.494
0.547
0.469
0.466
352.6
SLD [uu et a. 2024
8.741
0.348
0.436
0.435
0.463
375.0
Skeleton Difl. Currell et al.
2025
7.632
0.344
0.450
0.487
0.512
9.456
0.480
0.545
0.561
0.580
1013.5
CacheFlow
6.101
0.369
0,473
0.481
0.511
7.099
0.511
0.566
0.567
0.586
1.3
w/o Cache
5.385
0.374
0.489
0.490
0.531
6.291
0.516
0.586
0.573
0.608
415.9
performing 15 actions are recorded at 50 Hz. We follow the sctting including the datasct split, the 16-joints pose skeleton definition, and lengths of past and future motions proposed by previous works (Martinez el al.,
2017b; Luvizon et al. 2018 Yang et al. 2018 Pavllo et al., 2019). The training and test sets of Human3. 6.
are subjects [S1,55,S6 $6,5,,6,,T S7, and [S9,S11], respectively. The past motion and future motions contain 25 frames (0.5 sec) and 100 frames (2.0 sec). AMASS unifies 24 different human motion datasets including HumanEva-1 (Sigal et al. 2010) with the SMPT. (Loper et al. 2015) pose representation. AMASS contains 9M frames at 60 Hz in total. As a multi-dataset collection of AMASS, one can perform a cross-dataset evaluation. We follow the evaluation protocol proposed by BcLFusion (Barquero al., 2023) for fair comparison, as predicting future 120 frames (2.0 sec) with 30 frames observation (0.5 sec) with downsampling to 60 Hz.
Metrics. We use the evaluation metries to mcasure diversity and accuracy. 50 sampled predictions are evaluated with the following metries: Average Pairwise Distance (APD) AAiikbarrian et al.) 2020) evaluates sample diversity. It calculates the mean la distance between all predicted motions. Average and Final Displacement Error (ADE, FDE) (Alahi et al.] 2016 Lee et al. 2017 Gupta et al. 2018) evaluate accuracy. They calculate the average and fhal-framee l2 distances between the ground truth motion and closest prediction in the 50 set. Multimodal ADE and FDE (MMADE, MMFDE) (Yuan & Kitani 2019) also evaluate accuracy in a similar way to ADE and FDE. However, they are calculated over multimodal ground truths selected by grouping similar motions.
We also evaluate the accuracy of density estimation with Multimodal Log Probability per dimension. It calculates the log probability of the multimodal ground truths to measure how accurately the estimated density covers possible future motions. We evaluate the log probability on the motion space except for methods with latent space such as our CacheFlow and Bel.ou While higher is better on APD and multimodal log probability, lower is better on ADE, FDE, MMADE, and MMFDE.
Implementation Details. Our method is based on latent flow-base model as mentioned in Sec. 3.2 We utilize Variational Autoencoder (VAE) for D to obtain a latent representation. Specifically, we employ the Behavioral Latent Space (BLS) (Barquero et al., 2023) as a VAE to achieve a compact latent representation. BLS ensures smoothness of predicted motions and consistency between the end of the past motion and the start of the predicted motion. Additionally, we compress this representation using linear factorization (Xu et al. 2024). The dimensionality of the VAE latent space is 128, which we further reduce to 8 dimensions through linear factorization. We trained the unconditional flow-based model on this 8-dimensional space. The unconditional flow-based model fo is a continuous normalizing flow (CNF) model, with its vector
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Table 2: Density Estimation Accuracy on Human3.6M, Inference time each method reported as {total time (time without KDE inference)). Since our method doesn't 't require KDE for density estimation, the number of samples for KDE is left. blank for CacheFlow.
#sample for KDE
MM log prob. per dim f
Inference
Method
Time +
BeLFusion
Barquero et al. 2023
50
-2.383
2305.3
(440.3)
1000
-1.633
2422.4
(449.3)
CoMusion Sun & Chowdhary 2024
50
-15.575
2500.5
(167.0)
1000
-12.746
5071.5
5
(2741.3)
SLD Xu et al. 2024
50
0.080
2559.1
(375.0)
CacheFlow
1.304
0.5
(0.5)
field regressed by U-Net architecture. The conditional base density qá, as well the VAE encoder and decoder, are implemented as one-layer Recurrent Neural Networks (RNNs). We used Gaussian mixture model with M = 50 modes to model the conditional base density 4s. We precomputed and collected triplets f = (zz, (zt)) } using all training samples of cach datasct. All experiments, including inference time measuring, were carried out using a single NVIDIA A100 GPU. We used a batch size 64 and the Adam optimizer with a learning rate x 10-4.
5.1 Quantitative Evaluation
Accuracy Over a Fixed Number of Predictions. We compare CachoFlow against stato-of-tho-art methods of stochastic human motion prediction. While we propose using a precomputed set during inference, we also evaluate our method without precomputation. In the absence of precomputation, we sample from the conditional base density (z(z(c) (c) and obtain through the flow-based model inference, where æ = ==o(z). The results are summarized in Table I Since the primary applications of human motion prediction are in real-time scenarios, we also measure the inference time of each method to sample 50 predictions on a GPU.
CoMusion and SLD were successful in predicting motions that are closer to the ground truth than CacheFlow; however, their inference times of 167 and 375 milliseconds are too long for the intended 2000 ms prediction horizon. As result, over 8% of the first prediction sequence is rendered useless once the prediction is finalized. Therefore, it is difficult to use these methods with slow inference in rcal-time applications. Although our primary goal is to istim the densily, CacheFlow achieves comparable performances with 1.3 millisecond inference time. Our method achieves around 4x faster than the fastest VAE method, GSPS, and 30x faster than the fastest diffusion-based method, BeLFusion-D. The inference of our method is fast enough (1.3ms for future 2000ms) and applicable for real-time applications. This inference speed is because the inference of the unconditional flow-based model fo precomputed. We only need to evaluate the lightweight conditional base density 46 at inference. Although our conditional base density qá is just a Gaussian mixture with low expressive power, our method achieves high accuracy since the procomputed unconditional flow-based model Jo gives 46 much complexity with almost no overhead in inference.
Density Estimation Accuracy. The density estimation accuracy of each method is compared between CacheFlow and the otate-of-the-art methods. The three state-of-the-art methods BeLFusion (Barquero et al, 2023), CoMusion (Sun & Chowdhary 2024). and SL.D (Xu et al. 2024g are selected. CoMusion and SLD were selected since they outperform our mothod in benchmarks of stochastic human motion prediction. We also include BeLFusion to compare CacheFlow with the method with latent space. We applied KDE to these previous methods since they only sample set of predictions and cannot estimate density. While we evaluated 50 and 1000 samples for KDE on BeLFusion and CoMusion, SLD only allows 50 samples due to the fixed number of anchors corresponding to predictions. We measured the inference time of each method to estimate the density of ten thousand future motions from the past motion input.
The quantitative comparisons over the multimodal ground truth log probability are shown Tab. 2 All previous methods suffer from slow inference of their own and KDE on high-dimensional motion data. Their inference time exceeded the prediction horizon of 2000ms in the future. Therefore, they cannot estimate

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Published in Transactions on Machine Learning Research (01/2026)
Start
End Poow 10 sample
Start
GT
End Powe of sample
R
Del.
CacheFlon CoMmion
Farthest GT
Farthest
AMASS GRAB
AMASS DanceDB
Figure 3: Qualitative Comparison AMASS dataset.
Table 3: Zero-shot evaluation 3DPW (Von Marcard et al.,
2018 The models are trained on AMASS Mah-
mood et al. 2019 For CacheFlow, the cache are computed on AMASS
training set also.
Zero-shot
3DPW
(Von Marcard et al.
2018
APD
ADE
FDET MMADET MMFDEL
TPK Walker et al. 2017
9.582
0.648
0.701
0.665
0.702
DLow
Yuan & Kitani
2020
13.772
0.581
0.649
0.602
0.651
GSPS Mao et al.
2021
11.809
0.552
0.650
0.578
0.653
DivSamp Dang et al.
2022
24.153
0.554
0.678
0.593
0.686
BeLFusion
Barqqero
2023
7.740
0.493
0.590
0.531
0.599
CoMusion
Sun & Chowdhary
2024
11.404
0.477
0.570
0.540
0.587
SkeletonDiff. Curreli et al.
2025
9.814
0.472
0.575
0.535
0.594
CacheFlow
7.963
0.499
0.591
0.539
0.601
density in real-time. In contrast, our method achieves better estimation accuracy in less than one millisecond. This indicates that CacheFFlow has strong discriminative ability list up possible future motions required for safety assurance. Our method is even faster only the density estimation (0.5ms) than the inference time reported in Tab. [ (1.3ms). This is because we don't need any extra sampling operation in the density estimation.
5.2 Qualitative Comparison of Predicted Motions
visually evaluate CacheFlow, we conducted a qualitative comparison of methods on the AMASS dataset, as shown in Fig. 3 We visualized the end poses of 10 samples from each method alongside the end poses of past motions and the ground truth future motions. The sitting or lying poses were translated to the ground plane, the global translation is not modeled in human motion prediction. The 10 pose samples are arranged from the closest to the farthest from the ground truth pose based on joint rotations.
Our observations indicate that CacheFllow predicts realistic poses. The closest poses to the ground truths also demonstrate that the accuracy of CacheFlow is comparable to CoMusion, as reflected in the ADE and FDE metrics listed in Tab. ! Notably, our method is computationally efficient, operating 100 times faster than the fastest CoMusion. In summary, CacheFlow effectively delivers realistic and accurate predictions.
5.3
Further Analysis
5.3.1
Generalization to Out-of- Dff-siiiiibbtiin Data
To evaluate the generalization capability of CachePlow on unseen domains, we conduct a zero-shot evaluation on the 3DPW dataset (Von Marcard et al. 2018). In this setting, all models are trained strictly on the AMASS dataset (Mahmood al.) (219)) and tested directly on 3DPW without any fine-tuning. This poses a significant challenge due to the domain shift between the indoor optical motion capture data. of AMASS and the in-the-wild outdoor sequences of 3DPW.
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Table Computational and memory cost analysis on Human3. 6.6 and AMASS datasets.
Cost
Memory
Computation
Byles
Cache Size
GPU mem.
GPU
CPU
# Mot.
per Mot.
(MB)
(MB)
Cache
Inf.
Cache
Inf
Human3.6M
68
371188
24.07
811
26min 2.7ms
1h19min 3.6ms
AMASS
68
723263
46.90
875
1h10min 3.4ms
3h40min 6.9ms
The results are summarized in Tab. Despite the challenging zero-shot setting, CacheFlow demonstrates robust generalization performance. Specifically, method achieves an ADE of 0.499 and an FDE of 0.591, surpassing carlier basclines such as TPK, DLow, and GSPS. Furthermore, our performance is highly competitive with recent beafusion-bbsed approaches, such as BeLFusion (Barquero et al. 2023 (0.493 ADE) and CoMusion (Sun & Chowdhary 2024) (0.477 ADE), showing only marginal differences in error metrics.
A crucial aspect of this experiment the configuration of the cache. As noted in the table, the cache used by CacheFlow is constructed solely from the AMASS training set. The competitive performance on 3DPW indicates that CacheFlow does not simply memorize training samples. Instead, it successfully leverages the retrieved motion priors from the source domain (AMASS) to reconstruct valid motion trajectories in the target domain (3DPW). This suggests that the latent flow capabilities of CacheFlow effectively bridge the domain gap by generalizing the transitions found in the cached Training data to unseen, in-the-wild scenarios.
CacheFlow generalizes beyond training data in two key ways. First, although cached latents ã are drawn from the training set, generated futures X differ significantly because they are conditioned on novel past contexts c. This allows the model to generate motions unseen during training. Second, the large scale of the Human3.6M (371,188 patterns) and AMASS (723,263 patterns) datasets already provides considcrable diversily, and CacheFlow leverages this diversity to further expand to novel futures. Even with fixed cached codes, novel imputs yield generalized future motions that go beyond memorization of the training set.
5.3.2 Extensive Computational and Memory Cost Analysis
We conducted detailed evaluation of both computational overhead and memory requirements for CacheFlow across motion prediction benchmarks, using an Intel(R) Xeon(R) Gold 6154 CPU and single NVIDIA TITAN RTX GPU. The results demonstrate that our caching mechanism achieves efficient memory usage while maintaining practical inference times on both CPU and low-performance GPU platforms.
Memory Cost (MB)= = Bytes per motion x #motion / 11024
The measured costs are summarized in Table 4
This analysis highlights three important aspects. First, the cache footprint is very small, less than 50 MB for both datasets. Second, the preprocessing cost is modest, requiring only one to four hours even for AMASS on CPU. Third, the inference time highly practical, amounting only a few milliseconds per sample on both CPU and GPU, which makes CacheFlow suitable for real-time applications.
5.3.3 Ablation Study
We conducted an ablation study investigate how each component alfects the performance of our CacheFlow. We ablate five components: (1) dimensionality reduction via linear factorization VAE, (2) the unconditional flow-based model fo, (3) joint learning of the conditional base density 4s and unconditional flow-based model fe, (4) dataset for precomputation, (5) the sampling method for metrics over fixed number of predictions. Ablation results on the Human3.6N dataset are summarized Tab. 5
Lincar Factorization. We first ablate the linear factorization compressing 256-dim VAE latent be an 8-dim factor space. Our method is considerably enhanced on the compact space by avoiding the curse of dimensionality.
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Table 5: Ablation Study on Human3.6M. (4) and (5) do not affect the ground truth log probability, these are left
blank.
Linear
Unconditional
Joint
MM log prob.
Inference time/ms]
Precomp. Set
Sampling
ADEL
FDE
Factorization
Flow-based Model
Learning
per dimst
(1)
Train Set
NN sample
0.502
0.664
0.458
4.8
(2)
Train Set
NN sample
0.616
0.889
0.901
0.4
(3)
v
Train Set.
NN sample
0.370
0.475
1.283
1.3
(4)
Base Density
NN sample
0.376
0.492
1.3
Train Set
Random sample
0.455
0.605
1.2
(5)
/
Train Set
Most likely
0.384
0.506
1.4
/
Train Set
NN sample
0.369
0.473
1.304
1.3
The Unconditional Flow-based Model. We ablate this flow-based model So to confirm it improves the conditional base density qs by adding complexity. As shown in Tab. P we observe a notable performance drop without the flow-based model. Thercfore, our unconditional flow-based model fo complements conditional base densily qs to estimate complex density distribution over human motions.
Joint Learning. We ablate the joint learning of the unconditional flow-based model fe and the conditional base density 4s- The joint learning certainly improves both prediction errors and density estimation accuracy. The unconditional flow-based model fo can learn more clustered * mapped from the motion feature æ. Thus, a conditional base density de can casily model the z distribution.
Cache Precomputation. We propose the precomputation over the training split. Specifically, we apply inverse transform x fo( to ground truth future motions in the training split. However, we may precompute infinite procomputation samples. For example, we can sample
go(2|c) and obtain by forward transform æ = fo<)) As shown in the ablation, precomputation on the training split outperforms one on the base density since we can regularize the prediction to be legitimate human motions using the training split.
Sampling Method. We propose the ncarest neighbor sampling from the procomputation set as described in Sec. Lastly, we ablate this sampling lo evaluate its performance gain. We experimented with two sampling method alternatives: random sampling and most likely sampling. Precomputed motion features Zx- are uniformly selected as predictions with random sampling. Most likely sampling selects motion features æx We found that the large and little performance drops
with the highest probabilities A* = argmax,p(2* with random and most likely sampling respectively. This random sampling is worse due to the independence from the past motions c. The most likely method underperforms due to less diverse samples. It cannot select a motion feature set with diversity because all selected features are often located in one peak of the estimated density. Since ADE and FDE are best-of-many metrics, this less diversity leads to worse performance. In contrast, our sampling method superior to others. Our sampling incorporates past motions and achieves good diversity by simulating sampling from the estimated density p(ælcc).
6 Concluding Remarks
We presented a new flow-based stochastic human motion prediction method named CacheFlow. Our method achieves fast. and accurate estimation of the probability density distribution of future motions. Our unconditional formulation allows procomputation and caching of the flow-based model, thus omitting large portion of computational cost at inference. The unconditional flow-based model enhanced the expressivity of the lightweight conditional Gaussian mixture with almost no overhead. Experimental results demonstrated CacheFlow achieved comparable prediction accuracy with 1.3 milliscconds inference, much faster than the previous method. Furthermore, CacheFlow estimated a more accurate density than previous methods in less than 1 millisecond.
Our method has one limitation. Prediction and density estimation are performed within precomputed triplets. We cannot estimate the density or predict unseen future motions during precomputation. Our future work is searching for better precomputation strategy for prediction and estimation with more coverage based on the limited dataset. Furthermore, our method is not limited to prediction tasks but applies to any regression task requiring density estimation. We will investigate the applicability of our CacheFlow on other domains.
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A Reproducibility Statement
We are committed to ensuring the reproducibility of our results. To this end, we will release the full source code, including training and evaluation scripts. All datasets used in this study (Human3.6M) 6M (Ionescu et al.) 2013), AMASS (Mahmood et al., 2019), and derived benchmarks) are publicly available. Our implementation is based on PyTorch, and all experiments except Sec. 5.3.2 were conducted on a workstation with an Intel Xeon Gold 6154 CPU and NVIDIA A100 GPU.
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High
Probability
Low
Figure Visualization future motion densities by CacheFlow. The estimated densities for four different motion sequences are visualized. We used UMAP to project these future motions onto 2D space. Each dot represents an evaluated future motion, and the color of each dot indicates its probability, as shown in the side color bar. The red stars represent the projected multimodal ground truth future motions (Yuan & Kitani 2019). Multimodal ground truths are defined by grouping similar motions, which can be interpreted as multiple distinct possibilities.
We report detailed computational and memory costs in Table and we describe architectural and training hyperparameters in Appendix B. For reproducibility, (Adam with =0.9, =099.9 learning rate 5 x 10m4), batch sizes (192 for Human3.6M, 64 for AMASS). Random seeds were fixed to 0 across all experiments.
specify the number of training epochs, optimizer
B Implementation Details of Kernel Density Estimation
We asscssed the accuracy of density estimation using Kernel Density Estimation (KDE) previous methods. To ensure fair comparison of inference time, all KDE computations were conducted on the GPU. We applied KDE to the standardized predicted future motions (or latents for BeLFusion) to obtain the estimated density. In this process, the i-th dimension of the predicted future motions was standardized using its i-th variance, meaning that covariances were not considered during standardization. We employed Scott's rule to determine the optimal bandwidth for KDE.
C Visualization of Estimated Density
We visualized the future motion density estimated by CacheFlow. Since future motions are -igh-ddmeessionl data, we used UMAP (MeInnes & Healy 2018) to project each future motion into a 2D space. We displayed the multimodal ground truth future motions (Yuan & Kitani 2019) alongside the visualized density map. Multimodal ground truths are defined by grouping similar motions, which can be interpreted multiple
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Table Comparison to the GMM only model (without Flow Matching) on Human3.6M.
APD
ADE FDE MMADE MMFDE
CachePPlow 6.097 0.369 0.472 0.481
0.510
GMM only 4.545 0.548 0.751 0.637
0.763
Table 7: The APD (Average Pairwise Distance) and FID (Frechet Inception Distance) the previous methods and CacheFlow on Human3.6M.
APD
FID
GSPS
14.757
2.103
DivSamp
15.310
2.083
BeLFusion
7.602
0.209
HumanMAC
6.301
CoMusion
7.632
0.102
CacheFlow((Orr))
6.101
0.196
distinct possibilities. As shown in Fig. CacheFlow estimated high probability around the ground truth in all motion sequences. This visually supports the high density estimation accuracy presented in Tab. 2
D Comparison to GMM Only Method
While we propose a combination of a conditional Gaussian mixture model and unconditional flow matching, a Gaussian Mixture Model (Gll))only baseline might appear sufficient to model the conditional distribution. However, our experiments reveal that this is not the case. As shown in Tab. G this approach significantly degrades performance across all metries. While GMM itself can approximate conditional distributions, it lacks the expressive power to capture complex latent distributions effectively.
E Performance Gap on APD
While APD measures diversity via pairwise distances between generated samples, higher APD does not always imply better diversity, In fact, excessively high APD can indicate unrealistic or noisy predictions that ignore the input context.
For example, GSPS (Mao al. 2021 and DivSamp (Dang et al. 2022) report very high APDs (14.757 and 15.310), but we observed failure cases such as unrealistie floating motions, as also mentioned in other paper (Chen et al.] 2023) In contrast, recent diflusion-based models (BeLFusion (Barquero al. 2023
HumanMAC (Chen et al. 2023 CoMusion (Sun & Chowdhary 2024 prioritize plausibility and contextual
consistency, resulting in moderate APDs (7.602, 6.301, 7.632).
Our method achieves a comparable APD (6.101) to these diffusion-based methods, indicating a good balance between diversity and realism. Furthermore, we evaluated Frechet Inception Distance (FID), which better captures semantic and distributional similarity to real data. As shown Tab. 7 CacheFlow achieves significantly better FID than GSPS and DivSamp, supporting our claim that our method generates realistic and diverse future motions.
F
Cache Augmentation and Adaptation
CacheFlow is naturally suited for augmentation and adaptation. New motions can be added incrementally without retraining, simply by recomputing cached flows for the additional data. When transferring to a new dataset, cache recomputation provides fast adaptation and requires significantly fewer resources than retraining the full flow. Furthermore, since the cache acts as an external memory, continual learning becomes feasible without catastrophic forgetting. This property makes CacheFlow practical for long-term deployment in real-world scenarios such as assistive robotics and animation pipelines.
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G Potential Broader Impact
Research in human motion prediction, including our proposed CachePllw, carries significant societal implica- tions that must be carefully considered. Our primary goal is to advance computational understanding of human movement, enabling positive applications in fields like biomechanics, physical rehabilitation, robotics, and immersive virtual reality environments. However, we acknowledge the potential for misuse and the ethical challenges inherent in this technology.
G.1 Potential Positive Impacts
The proposed CacheFlow introduces a fast probaility-aware motion prediction framework, which may involve the following broader impacts:
Improved Collaboration in Robotics and Automation. In collaborative robotics and industrial automation, understanding and anticipating human motion critical for ensuring safety and efficiency. The proposed system enables robots to predict human actions and movements with probabilistic confidence, allowing them to adjust their trajectories and tasks in real time. This leads to smoother coordination in shared workspaces such as manufacturing floors, warehouses, or hospitals, where humans and robots must work in close proximity.
Proactive Support Assistive Technologies. assistive technologies for the elderly and indi- viduals with disabilities, anticipating human motion is essential for delivering timely and meaningful support. A fast and uncertainly-aware human motion prediction system enables robots and smart devices to proactively assist users by foreseeing movements such standing, walking, or reaching, even in the presence of noisy or partial sensor data. Furthermore, such system could help prevent falls or injuries by detecting signs of instability and initiating interventions early.
Immersive Interactions in VR and Gaming. Virtual reality (VR) and gaming systems stand benefit from predictive models that can estimate future body movements in real time with associated uncertainties. This capability allows VR applications to reduce latency and create more responsive environments by anticipating user actions and gestures.
G.2 Ethical Concerns and Mitigation Strategies
We recognize two main areas of ethical concern: privacy and potential misuse.
G.2.1 Privacy and Surveillance
Systems that predict human motion are often trained on public datasets (like AMASS and 3DPW) that contain recordings of real individuals. While these datasets typically anonymized by representing the subject skeletal mesh (e.g., SMPL model) without texture identity information, the technology itself could be integrated into rcal-time surveillance systems. The ability to predict future movement can enhance tracking and identification. As researchers, we strongly oppose the use of this technology for unauthorized surveillance or discriminatory profiling. Our research focus is strictly on foundational scientific advancement. and is confined to lab-controlled data and applications that respect user consent.
G.2.2 Bias and Fairness in Prediction
Bias in motion datasets is a recognized issue, as they may not equally represent all human populations across factors such as body shape, age, gender, culture, or ability. If CacheFlow were deployed on biased data, its predictive accuracy could be lower for underrepresented groups, potentially leading to critical failures in safety-critical applications (e.g., in advanced driver- assistance systems or elderly care robotics). We mitigate this by using large, diverse datasets (like AMASS) and demonstrating robust generalization data (3DPW), but we emphasize that continued dataset development focusing on fairness and inclusivity is critical for the field.
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G.2.3 Dual-Use Use Concerns
The core technological capability of anticipating human actions presents a dual-use challenge. While the positive applications are numerous, the technology could be leveraged to anticipate behavior in adversarial or unauthorized contexts. Our work is purely academic and aims contribute to the open body of knowledge for socially beneficial purposes. We encourage the responsible deployment of our method only in contexts with clear consent and explicit ethical guidelines.
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Reviews:
[{"review_id": "KFygmFD77e", "reviewer": "Reviewer_RoTQ", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "- This paper works on efficient and explicit future 3D human motion density estimation. The authors introduce a novel two-stage method combining an unconditional flow-based generative model with a lightweight conditional Gaussian mixture model.\n- This paper leverages a caching mechanism that stores flow-based latent-motion pairs, significantly accelerates the inference time without sacrificing the performance.\n- The proposed algorithm provides fast and likelihood-aware sampling without relying on kernel density estimation (KDE).", "claims_and_evidence_yn": "Yes", "claims_explanation": "The paper consists of several innovations (see above). However, there are certain concerns to be considered:\n\n1. Limited Generalization\nThis method relies on cached triplets from the training set and the prediction is constrained to this distribution. This may not generalize well to unseen motion patterns or out-of-distribution (OOD) events.\n\n2. Poor Diversity (APD) Compared to Some Prior Work\nThe APD metric is significantly lower than that of some sampling-heavy methods like DivSamp or GSPS. Although the authors argue that APD may not relate to the actual diversity, they didn't provide other metrics that can provide faithful evaluation.\n\n3. Limitation of the task\nThe proposed task is to estimate the \"density\" of future motion. However, this density significantly relies on the data distribution and is not a well-defined problem. Different datasets can yield to very different density estimation.", "audience_yn": "No", "audience_explanation": "I don't think the problem is a well-defined problem and I failed to see any important applications of this task.", "requested_changes": "Please provide more concrete evidences showing that this method can be used in real-world downstream applications.", "broader_impact_concerns": "I don't think the problem is a well-defined problem and I failed to see any important application of this task. Most real-world motion prediction tasks (human robot interaction, pedestrain behavior prediction, animation, etc.) contain more contextual background that can significantly change the density distribution.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "Pmvvqv6SSf", "reviewer": "Reviewer_g9rL", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper presents a new method of 3D human skeleton motion prediction and density estimation. The goal of this work is to accelerate the estimation of future skeletal frames $\\mathbf{x}$, given a history of past frames $\\mathbf{c}$, and accurate estimate of the probability density of future motion $p(\\mathbf{x}|\\mathbf{c})$. This is achieved by splitting the motion prediction problem into two steps. \n\nIn the first step, an unconditional motion model of future motions, $\\mathbf{x}$, is trained based on a training set. A conditional normalizing flow model is used here, and the source noise distribution is a simple Gaussian distribution. Then triplets of $(\\mathbf{z},|det(J(\\mathbf{z})|^{-1}, \\mathbf{x})$ for future motions in the training set are cached. \n\nIn the second step, a lightweight conditional generative model is built, where the conditioning input is the past motion and the conditional base distribution $q(z|c)$ is a small learnable model. Once $q(\\mathbf{z}|\\mathbf{c})$ can be computed, computing $p(\\mathbf{x}|\\mathbf{c})$ is easy through $p(\\mathbf{x}|\\mathbf{c}) = q(\\mathbf{z}|\\mathbf{c}) |\\text{det} J(\\mathbf{z})|^{-1}$. The authors choose to model $q(\\mathbf{z}|\\mathbf{c})$ as a Gaussian mixture model (GMM) with $M$ Gaussians. Given a condition $\\mathbf{c}$, the means and variance of the $M$ Gaussians are regressed using a single-layer RNN. \n\nAt test time, given a condition $\\mathbf{c}$, the RNN produces the GMM components. $q(\\mathbf{z}|\\mathbf{c})$ is obtained using the z’s from the cache and evaluating the probability under the GMM prior, and then $p(\\mathbf{x}|\\mathbf{c})$ is computed using the \n\nBoth the unconditional motion flow model and the lightweight conditional generative model are trained jointly. \n\nThis simple two-step process makes the inference process 4x to 30x faster compared to existing methods, without any significant drop in human motion prediction performance on standard datasets.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The two main claims are that with this simplified model of human motion prediction, performance does not drop too much and the inference speed is much higher. Table 1 on two common datasets, Human3.6M and  AMASS, show promising results in terms of motion prediction accuracy, diversity and density estimation. Inference time is about 4x faster compared to the GSPS method, with similar errors, and about 30x faster than diffusion-based models. \n\nThe ablation results in the appendix show the importance of various parts of the pipeline. \n\nThe authors also say that code for this paper will be released.\n\nBased on these factors, I believe the claims in the paper are supported well by the experimental results.", "audience_yn": "Yes", "audience_explanation": "Yes, the findings in this paper would be interesting to some individuals in TMLR’s audience. Motion modeling and prediction is more general and useful in other domains like robotics, where similar methods could be employed, where fast inference is very important. \n\nAlso, the fundamental idea in this paper of storing training set latents and performing a simple smart lookup at test time is shown to get good performance in human motion prediction, I believe this is interesting.", "requested_changes": "The main issue with the paper is the writing and explanation of concepts. I have listed a few changes that I think can improve the paper’s presentation.\n\n1. Unclear notation: Many symbols in the paper are confusing to me.\n\nOne issue is with $\\mathcal{E}$. I think this is referring to an encoder network. But I don’t understand where this fits in the rest of the pipeline. Do you train an encoder-decoder on the skeletons first and then work only on the latent representations? I think this is explained partly in Section 3.2. \n\nDoes this mean there are two flow models in the paper? One for encoding sequences and the other for mapping from $\\mathbf{z}$-space to $\\mathbf{x}$-space? Is the BLS model this initial encoding model? \n\nAre the conditioning inputs, $\\mathbf{c}$, not encoded?   \n\nWhat is the unconditional base density $q(\\mathbf{z})$? Is it a Gaussian?  \n\nIs $\\phi$ supposed to denote the parameters in the RNN?  In the line before Eq (10), its says “… from a source distribution $q_\\phi(\\mathbf{z}|\\mathbf{c})$”. This should be $f_\\theta(\\mathbf{z})$, right?  \n\nMore clarity from the authors is needed here. \n\n2. I don’t understand the point moving many important results, like the ablation study, to the appendix. I don’t think TMLR has any paper length limits, I think pretty much all of the appendix should be moved to the main paper.  \n\n3. I don’t understand Figure 4 fully. Why is there more than one red dot for ground-truth future motion? This is confusing to me. I hope the authors can improve the explanation here.", "broader_impact_concerns": "No concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "AOIzO2pQgQ", "reviewer": "Reviewer_7ULe", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "CacheFlow proposes a novel two-stage approach for fast stochastic 3D human motion prediction that combines unconditional normalizing flows with a lightweight conditional mapping. Instead of learning a full conditional flow (expensive) the authors precompute and cache results from an unconditional continuous normalizing flow (CNF), then train a simple Gaussian Mixture Model conditioned on past motion to link to these cached results. This design allows the model to bypass heavy flow computation at inference, achieving faster prediction time, compared to VAE-based methods and diffusion models, while maintaining comparable accuracy on Human3.6M and AMASS datasets. The paper demonstrates that CacheFlow delivers accurate probabilistic density estimation, making it suitable for real-time human motion forecasting in robotics and animation.\n\n**Strengths**:\n- Major Efficiency Innovation\n\n**Weaknesses**\n- Narrow Experimental Scope: Evaluations are restricted to Human3.6M and AMASS; missing tests on more diverse or in-the-wild datasets (e.g., 3DPW, GRAB, KIT-ML) limits claims about generalization and robustness.\n- Incomplete Literature Coverage: The related work omits several important recent advances, especially diffusion-based motion models (e.g., MDM, MotionGPT, MotionLM) and flow acceleration techniques (e.g., Rectified Flow,), making the context appear slightly dated.", "claims_and_evidence_yn": "No", "claims_explanation": "Claim: CacheFlow achieves substantially faster inference (about 1 ms). **Supported**. Table 1 and 2 show detailed runtime comparisons against GAN, VAE, and diffusion baselines (e.g., 1.3 ms vs 39 ms for BeLFusion-D, 352 ms for CoMusion).\n\nClaim: Prediction accuracy is comparable to state-of-the-art methods. **Partially Supported**. Results on Human3.6M and AMASS report ADE/FDE metrics worse than diffusion-based models.\n\nClaim: CacheFlow improves explicit density estimation accuracy. **Convincingly supported**. Table 2 compares multimodal log probability per dimension with baselines using KDE. CacheFlow achieves the best (higher) log-probability with orders-of-magnitude lower computation.\n\nClaim: Real-time applicability for robotics or safety-critical systems. **Plausible but speculative.** The paper provides no actual robotics or real-world deployment tests; only simulation benchmarks.", "audience_yn": "Yes", "audience_explanation": "CacheFlow’s contribution (computationally efficient, explicit probabilistic modeling for high-dimensional motion data) aligns directly with the interests of:\n\n- Generative modeling researchers (especially those studying flows, diffusion, or VAEs).\n\n- Human motion prediction and trajectory forecasting communities.\n\n- Applied ML engineers focused on real-time inference for robotics, AR/VR, and animation.", "requested_changes": "- Please fix the citation format. Many citations should be generated using `\\citep` rather than `\\cite` or `\\citet`.\n- Please include more recent works, outlined in the **Weaknesses** section above.\n- Please consider adding at least one other dataset in your evaluation pipeline.", "broader_impact_concerns": "There are obvious ethical concerns with the whole field of human motion detection. I encourage the authors to discuss them.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)