Hyperspherical Latents Improve Continuous-Token Autoregressive Generation
Authors: Guolin Ke
HUI XUE
Venue: ICLR 2026 Poster (2026)
Keywords: autoregressive generation
image generation
diffusion
Primary area: generative models
TL;DR: SphereAR improves continuous-token autoregressive generation by enforcing scale-invariant latents: all inputs and outputs—including post-CFG—are projected onto a fixed-radius hypersphere.

Abstract: Autoregressive (AR) models are promising for image generation, yet continuous-token AR variants often trail latent diffusion and masked-generation models. The core issue is heterogeneous variance in VAE latents, which is amplified during AR decoding, especially under classifier-free guidance (CFG), and can cause variance collapse. We propose SphereAR to address this issue. Its core design is to constrain all AR inputs and outputs---including after CFG---to lie on a fixed-radius hypersphere (constant $\ell_2$ norm), leveraging hyperspherical VAEs.  Our theoretical analysis shows that hyperspherical constraint removes the scale component (the primary cause of variance collapse), thereby stabilizing AR decoding. Empirically, on ImageNet generation, SphereAR-H (943M) sets a new state of the art for AR models, achieving FID 1.34. Even at smaller scales, SphereAR-L (479M) reaches FID 1.54 and SphereAR-B (208M) reaches 1.92, matching or surpassing much larger baselines such as MAR-H (943M, 1.55) and VAR-d30 (2B, 1.92).
To our knowledge, this is the first time a pure next-token AR image generator with raster order surpasses diffusion and masked-generation models at comparable parameter scales.

Published as a conference paper at ICLR 2026
HYPERSPHERICAL LATENTS IMPROVE
CONTINUOUS- TOKEN AUTOREGRESSIVE GENERATION
Guolin Ke
Hui Xue
DP Technology kegl @ dp.tech
Microsoft Research
xuehui @ wwcc@@@octccoooomommmmmoo
Code: httppp://gthhaa..ccmmmooo//////s
ABSTRACT
Autoregressive (AR) models are promising for image generation, yet continuous- token AR variants often trail latent diffusion and core issue heterogencous variance in VAE latents, which is amplified during AR decoding, especially under classifier-free guidance (CFG), and can cause variance collapse. We propose SphereAR to address this issue. Its core design is to con- strain all AR inputs and outputs including after CFG- lie fixed-radius hypersphere (constant & norm), leveraging hyperspherical VAEs. Our theoretical analysis shows that hyperspherical constraint removes the scale component (the pri- mary cause of variance collapse), thereby stabilizing AR decoding. Empirically, on ImageNet generation, SphereAR-H (943M) sets new state of the for AR mod- els, achieving FIDD 1.34. Even at smaller scales, SphereAR-I. (479M) reaches FID 1.54 and SphereAR-B (208M) reaches 1.92, matching or surpassing much larger baselines such as MAR-H (943M, 1.55) and VAR-d30 (2B, 1.92). To our knowl- edge, this is the first time a pure next-token AR image generator with raster order surpasses diffusion and masked-generation models at comparable parameter scales.
models. The
2.6
GIVT
2.4
FID (lower better)
LlamaGen
2.2
LatentLM
RandAR
2.0
SiT
VAR
1.8
1.6
MAR
1.4
SphereAR (ours)
300M
600M
18
28
Parameters (Millions), log-scale
Figure 1: Left: FID vs. parameters on ImageNet 256x256 conditiona generation, SphereAR attains lower FID with fewer parameters, Right: 256x256 samples generated by SphereAR-L(479M)
INTRODUCTION
Autoregressive (AR) models have achieved remarkable success text (Radford et al. 2018 Brown
et al. 2020) and have been extended to images (Esser et al.) [021]] Yu et al. (2021), speech (Meng
ctt al. 2024), video (Teng et al. 2025), and other modalities (Lu et al.) 2025). Early multimodal
AR systems discretized latents with vector quantization (VQ) (Gray) 1984 Van Den Oord et al.
2017); more recently, continuous token AR dispenses with codebooks: a VAE (Kingma and Welling
2013) emits token-level latents and the AR model predicts the next latent in continuous space (e.g..
Gaussian mixtures (Tschannen et al. 2024) or diffusion objectives (Li et al., 2024a Sun et al. 2) Yet, when built the same VAE latents, continuous-token AR models often trail latent

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Token
Token 2
Token
Encoder
CLS
L2 Norm
mean direction
K concentration
Autoregressive Transformer
with Diffusion Head
vMF/
Hyperspherical
Power Spherical
Latent Tokens
Decoder
X
Token
Token
Token
Figure 2: Overview of SphereAR. Left: A hyperspherical VAE (S-VAE) encodes raw data into a sequence of latent tokens constrained fixed-radius hypersphere gd-1, The encoder outputs unit mean direction and concentration that parameterize Power Spherical posterior. Right: A causal Transformer with token-level diffusion head models the next-token distribution over the hyperspherical token sequence. At inference, the AR model's predictions, including CFG-rescaled ones, are projected back onto the fixed-radius hypersphere. The VAE decoder then reconstructs the image from the predicted hyperspherical latents.
Mises Fisher (vMF) or
diffusion and sennkeddenneationn models Prior analyses attribute this gap variance pathologies during AR decoding (Sun et al., 2024b] Team et al. 2025): latent variances are heterogencous across dimensions/tokens and are amplified due to exposure bias and classifier-free guidance (CFG) (Ho and Salimans (202)), causing stepwise variance drift and collapse. Strengthening the KI. term (Tchannen al. (2024) or fixing a large variance (Sun et al., (024b) improves stability but leaves the root cause intact: scale heterogeneity remains and can still drift during AR decoding with CFG.
We address this with more principled solution: make all AR inputs and outputs scale-inven iinea-iivariant.
As illustrated in Fig. 2 the proposed SphereAR couples a hyperspherical VAE (S VAE) (Davidson
al., 22188 De Cao and Aziz (2200) with autoregressive Transformer (Vaswani et al., (017) and token-level diffusion head (Li et al. 2024a). The S- S-VAE constrains each latent token to a fixcd-radius hypersphere (constant le norm), parameterizing only direction via unit mean direction vector μ and scalar concentration x. During training, the AR model consumes these hyperspherical latents under teacher forcing. During inference, AR model's projected back onto the fixed-radius hypersphere to remove the radial (scale) component. Thus, every signal provided to or produced by the AR model is fz-normalized to the same radius. concise theoretical justification supports these design choices, showing why scale-invarian inputs/outputs stabilize AR decoding and why hyperspherical posterior is preferable to Gaussian alternatives.
those after CFG rescaling-
new state of the art for AR models ImageNet 256x256 class-conditional generation, achieving FID 1.34. Even at smaller scales, Sphere (479M) attains FID 1.54, outperforming comparably sized diffusion (DiT-XL/2, FID 2.27) and bidirectional masked- generation (MAR-L, FID 1.78) baselines, while matching MAR-H (943M, FID 1.55) with roughly half the parameters. At the base scale, SphereAR-B (208M) achieves FID 1.92, surpassing VAR-d20 (600M, FID 2.57) and the prior continuous-token AR model LatentLM-L (479M, FID 2.24), while matching VAR-d30 (2B, FID 1.92) with 100 fewer parameters. Ablations show that AR models with hyperspherical VAEs consistently outperform diagonal-Gaussen (Sun et al. 2024b) baselines; moreover, applying post-hoe normalization to diagonal- helps but still underperforms S-VAE. To our knowledge, this is the first time pure next-token AR image generator with raster order surpasses diffusion and masked- geneerrtnnrrtionn models at comparable parameter scales.
Empirically, SphereAR-H (943M)
and Exed-variance v-VAE
latents
In this paper, "autoregressive" denotes token- token-by-tole bo-token generation with unidirectional (causal) self-attention, excluding bidirectional masked/next-scal methods such as MaskGIT (Chang et al. 2022). MAR (Li et al. 2024a), and VAR (Tian al. 2024). With the same VAF latents, [L et al. (204a report AR model at FID 4.69 vs. 1.98 for MAR and 2.27 for DiT (Pechles and Xie 2023
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2 RELATED WORK
Image Tokenizers A large body of work improves the performance of image tokenizers by enhanc- ing reconstruction fidelity and semantic alignment. Typical ingredients include (i) refined training objectives (Esser et al., 202T; Yao et al. 2025; Yang et al., 2025), (ii) CLIP-aligned distillation for better text guidance (Peng et al. 2022; Qu et al., 2025). and (iii) various decoder improve- ments (Chen et al. 2025 Yang ct al. 2025). These techniques are orthogonal to our approach and can be combined with In parallel, complementary line of work targets the quantization mechanism itself- inelo-ii codebook utilization, training stability, and the rate-distortion tradeoff. Building on VQ-VAE (Van Den Oord et al. 2017), extensions include hierarchical VQ-VAE-2 (Razavi et al. (019)), residual/hierarchica quantization (Lee et al. 2022), and multi-codebook schemes (Li et al. (024b).
Some methods also adopt spherical or normalized feature geometry in the quantizer: for instance,
ViT-VOGAN Yu et al. 2021) normalizes latent features before computing codebook distances, and
BSQ (Zhao et al. 2024) constructs binarized spherical latents for bit-efficient quantization.
By contrast, comparatively less work targets continuous image tokenizers tailored to autoregressive modeling. Most prior approaches follow latent diffusion practice (Peebles and Xie 2023) and employ Saagonal-Gaussiar VAEs. GIVT (Tschannen et al. 2024) and LatentLM (Sun et al. 2024b) mitigate instability by inflating or fixing latent variance (e.g.. B V-V o-VAE), which helps but does not remove scale degrees of freedom. NextStep-1 I (Team et al. 2025) instead normalizes Gaussian- posterior latents to a constant norm, achieving scale invariance. However, both our theoretical analysis and empirical results indicate that hyperspherical posteriors are preferable to post-hoe normalization of diagonal-Gaus Gaussian latents.
Autoregressive Image Generation Autoregressive image generation can be grouped into three
families: next-scale, next-set, and next-token prediction. In next-scale prediction (e.g.. VAR (Tian) ct al. 2024)), images are generated coarse-to-fi across resolutions; within each scale, context is modeled bidirectionally. In next-set prediction (also called masked generation; e-g., MaskGIT (Chang ct al. 2022). MAR (Li et al. 2024a)), a single scale used and a sei tokens is updated in parallel under bidirectional attention. In next-token prediction (c.g., VQGAN (Esser et al.] 2021). LlamaGen (Sun et al.] 2024a)), the model follows language-style sequence modeling: one token is predicted at a time with strictly unidirectional (causal) attention. We focus next-token models because they align naturally with autoregressive language modeling and offer headroom for unified multimodal models.
A wide range of next-token variants has been explored: discrete tokens (Esser et al. 202 1 Yu
jet al. 2021; Sun et al. (024a) vs. continuous tokens (Tschannen et al. [024]] Sun et al. (024b)) raster order (Sun al.
2024a Tschannen et al.
2024) vs. randomized order (Pang et al. 2025 Yu
et al. 2024); and more (Li et al. 2025). However, at comparable parameter scales, these models
have often trailed next-set and next-scale approaches. A key reason the variance collapse that emerges during autoregressive decoding (Sun et al., 2024b] Team et al.) 2025). We address this by enforcing iiveriin latents via hyperspherical VAEs, thereby removing scale degrees of freedom. Empirically, this yields substantial gains for sequential AR decoding, with performance that matches or surpasses state-of-the-an next-set and next-scale methods at comparable model budgets.
3 METHOD
We observe that with discrete tokens, next-token autoregressive (AR) models can outperform bidi- rectional masked-generation (MG) approaches. For example, LlamaGen-L (343M, FID 3.07; (Sun
ct al. 2024ah) vs. MaskGIT (207M. FID 4.02; (Chang et al.) (0022). A system-level study (Fan et al. 2024) further reports that, with discrete tokens, AR consistently achieves better FID than MG across model sizes from 166M 3.1B. By contrast, with continuous tokens, MG is consistently stronger than AR. This divergence discrete tokens thriving under AR while continuous tokens do
not-motivates us to probe what truly differentiates the two. As illustrated in Fig 3 discrete tokens (Fig 3 a) are normalized on the probability simplex (components sum to 1), yielding scale-invariant inputs and outputs that stabilize AR decoding. In contrast, diagonal-Gaune Gaussian latents (Fig(3 b) are unconstrained and can destabilize multi-step AR decoding due scale drift that compounds across steps. We hypothesize this scale sensitivity is the key issue, and therefore constrain continuous latents to fixed-radius hypersphere to enforce a constant norm (Fig This idea underpins SphereAR:
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15
60
0.0
crss
Blenn
dass 5
coso
Token
Cass_
(a) Discrete
(b) Diagonal-Gaussian Gaussian
(c) Hyperspherical (constant & norm)
(probability simplex; -
(unconstrained)
Figure 3: Visualization of token distributions. Each panel shows one token type, with three tokens in different colors. (a) Discrete tokens lie on the probability simplex and are intrinsically scale-invariant. (b) Diagonal-Gaussian latents are unconstrained scale; despite a KL prior, per-dimension/tokee variances remain heterogeneous. (c) Hyperspherical latents constrain each token to fixed norm (e.g.. | = R), yielding iaaleiinvrriant representations. In practice, (a) and (c) are robust under AR decoding. whereas (b) is prone to scale drift and occasional variance collapse (e.g., with CFG).
hyperspherical VAE paired with causal Transformer equipped with token-level diffusion head; we detail these components below.
3.1 FROM VAE TO HYPERSPHERICAL VAE
variational autoencoder (VAE) (Kingma and Welling 2013) is widely used compress raw data
into lower-dimensional latent vector. It consists of an encoder (ø(z x) that parameterizes an
approximate posterior over z and a decoder Py: (x z) that reconstructs x from z. We train the model by maximizing the evidence lower bound (ELBO);
C(ó, 1o; x) = (x)())
(x z)] - DKI(go(z x) p(z)).
(1)
By default, both the prior p(z) and the approximate posterior (z x) parameterized as Gaussians with diagonal covariance; the prior is the isotropic standard Normal A (0, I). Using the reparameterization trick, z (x(x) + as (x)
with e (N I), makes the sampling step differentiable so that gradients backpropagate from the decoder to the encoder.
With this diagonal-Gausse posterior, the encoder's scale o (x) is data-dependent and per-dimension, yielding heterogeneous variances across dimensions and tokens. This imbalance amplifies exposure bias and can trigger variance collapse in AR decoding, particularly under CFG (Sun al.] 2024b]
Team et al. 2025)
Hyperspherical VAE (S-VAE) To fully address this issue, we remove the scale degree of freedom in the latent representation, rendering the AR model's inputs and outputs scale-invariant. Specifically, leveraging hyperspherical VAES (S-VABS) (Davidson et al. 2018: De Cao and Aziz. 2020), we
constrain each latent token to lie fixed-radius hypersphere.
For each token, the S-VAE encoder parameterizes a directional posterior on the unit sphere by outputting unit mean direction (x(x) E gd-1 (via & normalization; d the latent dimension) and a nonnegative concentration (x) E R>0- For notational convenience. let (x( and
(x). S- VAE adopts a von Mises -isher (vMF) distribution (Davidson et al. 2018 for the directional approximate posterior:
Is (u x) = Cd(k) exp(N TT)..
8d-1
where Ca(k) (2m) 1g (k)
the normalizing constant and (.(.) the modified Bessel function
of the first kind. Intuitively, je sets the preferred direction and controls concentration: gives
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the uniform distribution S and larger yields tighter mass around μ.. Because f is the cosine similarity on the sphere, the density in equation increases as u aligns with μ.
We take the prior directions to be uniform on the sphere, p(u) Unif($d-1), and use fixed radius > 0 (hyperparameter) so that z = R u fed to the decoder. The ELBO becomes
Ls.vAE(O, y x) (uj(() [log (x z Ru)] D x) p(u)).
While vMF is principled for spherical latents, it can be less efficient due to the need for rejection sampling. For efficiency, we adopt the Power Spherical posterior (De Cao and Aziz 2020 on g
9 x) (1+H²u) u)"
E u
which preserves spherical support and rotational symmetry yet admits fully reparameterizable sampler without rejection sampling. For convenience, define the axial projection (cosine similarity) f E [-1, 1] with the affine transform C' (c+ 1)/2 E [0, 1]. Under equation 4 the marginal of C is a Beta distribution with parameters determined by and s:
C~ Beta a d-1 -3)==))) 1
so that
c= 2C
(5)
Sampling proceeds by drawing C from the Beta and setting 2C - 1, then sampling unit vector Vi uniformly in the tangent space orthogonal to and composing
cp + V1 c2 V1,
(6)
optionally implemented via a Householder transform to align a reference basis with μr. This inverse- CDF construction yields low-variance, fully reparameterizable gradients and improved numerical stability; the spherical ELBO in equation 3 remains unchanged with taken as Power Spherical. In downstream autoregressive models, we keep the radius fixed and renormalize latent inputs/outputs back to ||z||2 = R (also after CFG rescaling) to remove scale degrees of freedom.
Why Scale- Invariant Inputs and Outputs Matter AR We normalize each provisional next- token prediction by the radius-R R. projection [z( Rz/lel |z|2 onto the hypersphere. At a reference token on the sphere, the differential of Nn is exactly the orthogonal projector onto the tangent space; thus, to first order, normalization removes radial (scale) perturbations and preserves only tangential (directional) ones. Consequently, composing normalization with the next-token predictor removes the radial (scale) component of the linearized one-step error prior refeeding, scale errors cannot accumulate across autoregressive steps. See Appendix A for the formal statement and proof.
Limitations of Gaussian Posterior with Post-hoc Normalization tempting alternative to achieve scale invariance is to retain (via NR), before feeding them to the decoder (henceforth "Gaussian+norm"), However, this choice is theoretically suboptimal: it optimizes a strictly looser variational bound than spherical posterior (see Appendix B). Intuitively, the decoder discards radius by normalization, yet the ELBO still incurs an extra nonnegative radial KL term that does not arise with hyperspherical posterior. By contrast, a hyperspherical posterior aligns the training objective with the constant- normm constraint and avoids this mismatch. Moreover, hyperspherical posteriors a single concentration parameter K, whereas Gaussian+norm induces a projected-normal (Angular Central Gaussian) directional law whose level sets are elliptical and generally not axially symmetric; this geometric mismatch makes it a poorer fit to purely directional structure (details in Appendix B). Empirically (Sec. 4.3) S-VAE outperforms Gaussian+norm. corroborating this analysis.
GGasssinn posterior qo x) and normalize the sampled latents
axially symmetric about ja and governed by
3.2 CONTINUOUS- TOKEN AUTOREGRESSIVE TRANSFORMER
Given an image X E I xxxx3 S-VAE encodes it into latent tensor Z E 1gxx with fixed per-token nor:: for every spatial location (i.j). || |Za, ll2 = R (cach Z E Rd). For sequen- tial autoregressive modeling,
flatten Z in raster-scan (row-major) order obtain sequence
{z]...., 21} of length / = h w, where 2ς is simply the latent at the k-th position in row-major order.
We employ causal (unidirectional) Transformer to model the conditional distribution of the next token the flat sequence. At position 1, the model takes the prefix [Z)..... Zg-1} as input and produces hidden state hg-1 = f(ack; 0), where 0 denotes the Transformer parameters. Optionally,
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discrete class labels or text prompts are prepended conditioning tokens to the prefix and included in the causal context.
To predict the next continuous token zy. follow MAR (Li ct al diffusion head. Conditioned on h the head progressively transtorms a simple prior (c.g., (0, (). into the data distribution of the next token Z₁.-
2024a) and attach token-level
We train the diffusion head with Rectified Flow Lipman et al. 2022 Liu et al. 2022). Given a prior
z (0, I),, target z Zy: and continuous time t (0, 1). we form the linear interpolation
z (1 (1-t) z + tzz.
The diffusion head, parameterized by w. takes the noisy interpolation zi. the scalar time I, and the condition h inputs, and predicts a velocity, vu(zi i, h&-1) Rd. The training target is the flow velocity along the straight path, di
dzf 2/ 2 and the objective is mean-squared emror:
RRF = Eng 1t, [1|2] - 2x - vulzko L, ha-
(8)
At inference, we initialize 2 ~ (N., at = and integrate the learned velocity field Vu (z), E, 11-1) to to
using N uniform steps A I/N (e.g., explicit Euler):
2 + z + Ava t, hg-1).
(9)
After N steps, we enforce the constant-norm constraint with a single projection onto the radius- R hypersphere: + R /l ll2. The resulting token Zξ is then fed to the next AR step and ultimately to the VAE decoder. When using classifier-free guidance (CFG), the velocity at each step is obtained from guided (rescaled) combination of conditional and unconditional predictions; we perform no intermediate normalization and apply single constant-norn projection only after N steps.
3.3 MODEL ARCHITECTURES
S-VAE Although VQGAN-style CNN backbones (Esser et al., 202T are effective for latent VAËs, their throughput is limited by large convolutional activation maps. To improve efficiency without sacrificing quality, we adopt a hybrid backbone: the encoder uses a lightweight CNN stem with downsampling for patchification, followed by stack of Transformer blocks; the decoder mirrors this with a Transformer stack that refines latent tokens and lightweight CNN with upsampling for umpatchification and pixel reconstruction. This preserves the CNN's strong local inductive bias while leveraging the Transformer's 's efficient global modeling at token resolution, yielding a favorable speed quuality trade-off. As shown in Appendix D. the hybrid matches CNN baselines in quality while being about 2.6x faster.
Autoregressive Transformer Following prior work (Sun al. 2024a b), adopt modern causal Transformer. Concretely, we use pre-norm Transformer blocks with RMSNorm (Zhang and Sennrich 2019; Xiong et al. 2020). Attenton for efficient attention computation (Dao al. 2022), and SwiGLU forward layers (Shazeer 2020). For image positional encoding, we employ 2D rotary embeddings (RoPE) (Su al.) (2224) applied in raster-scan order. All self-attention is
strictly unidirectional (causal mask). For the diffusion head, we follow MAR (Li et al. 2024a) and
use an MI.P architecture.
EXPERIMENTS
We evaluate SphereAR on ImageNet-1K (Deng al., 2009) class-conditional generation of resolu- tion of 256x256, comparing against previous strong baselines. Beyond end-to-end comparisons, we include targeted studies to substantiate our design choices, focusing on the following questions: (1) S- VAE vs. diagonal-Gaussian Does S- V-VAE outperform diagonal-Gaussian VAEs for continuous-token AR? (2) Post-hoc normalization: we apply l normalization to latents from a diagonal Gaussian VAE, how does compare with S- S-VAE? (3) Component contributions: Which parts of S-VAE drive the gains-(i (i) the hyperspherical posterior, (ii) normalization applied to the VAE decoder input, or (iii) normalization applied to AR inputs/outputs?
diagonal-Gaussian

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4.1 IMPLEMENTATION DETAILS
S-VAE We adopt Power Spherical De Cao and Aziz 2020) directional posterior with latent
dimensionality d = 16 and fix the radius to R vd. Complete setting for the S-VAE's backbone is provided in Appendix D We train S-VAE from scratch on ImageNet-1K (Deng et al. 2009)
with random-crop augmentation, optimizing weighted sum of ELBO (reconstruction + KL), perceptual (Johnson et al.) 2016: Zhang et al.] 2018), and adversarial (Isola et al.] (017) losses. Optimization uses AdamW (Kingma 2014 Loshchilov and Hutter, 2017) for 100 epochs (batch size 256, learning rate x 10 (0.9, 0.95), weight decay 0.05).
Autoregressive Transformer Following MAR (Li et al. 2024a), instantiate three model sizes for SphereAR. SphereAR-B uses 24 Transformer blocks (hidden size 768) and a diffusion head with 6 feed-forward blocks (hidden size 768). SphereAR-I. uses 32 Transformer blocks (hidden size 1024) and a diffusion head with 8 feed-forward blocks (hidden size 1024). SphereAR-IF H uses 40 Transformer blocks (hidden size 1280) and a diffusion head with 12 feed-forward blocks (hidden size 1280). As in MAR, we employ multiple class-conditioning tokens (16 in our models vs. 64 in MAR) and apply class-token dropout with probability 0.1 during training to enable classifier free guidance (CFG) at inference. Models are trained on ImageNet- IK for 400 epochs using S VAE latents with Adam W (batch size 512, 3 = (0.9, 0.95), weight decay 0.05), a cosine learning- schedule with 20k linear warmup steps and peak learning rate 3 10-1, and an exponential moving average (EMA) of weights with decay 0.9999. Under these settings, SphereAR-B SphereAR L and SphereAR-H H contain 208M, 4479M and ~943M parameters, respectively.
Inference Settings For next-token prediction we integrate the learned velocity field with a fixed- step Euler scheme (100 steps). We use the linear CFG schedule from MAR. We enable a KV cache to improve autoregressive decoding efficiency.
4.2 IMAGE GENERATION RESULT
We report Fréchet Inception Distance (FID) (Heusel al. (017) as the primary metric, computed on
50k samples drawn with a fixed random seed using the ADM evaluation code (Dhariwal and Nichol
2021). The optimal CFG scale is determined through a sweep with a step size of 0.1. Following MAR (Li et al. 2024a), we additionally report Inception Score (IS) (Salimans et al.) 2016) and
Precision/Recall (Pre./Rec.) (Kynkäänniemi et al., 2019].
From the results summarized in Table I we observe: (1) SphereAR-H (943M) achieves state-of-
the-art FID 1.34, outperforming VAR-d30 (2B, 1.92) and MAR-H (943M, 1.55). (2) SphereAR is
aarameter-efficeent At large scale, SphereAR- (479M) matches MAR-H (943M, 1.55) with roughly half the parameters. Even at the base scale, SphereAR-B (208M) reaches FID 1.92, outperforming 2B 1B-parameter VAR, diffusion basclines (DiT and SiT), prior continuous-token AR models (GIVT and M--en and larger discrete AR models (LlamaCien and RandAR). (3) Hyperspherical latents are critical. The key difference from LatentLM is the latent parameterization
fixed-varianee diagonal-Gaussin vs. hyperspherical. The large gap (SphereAR-L: 1.54 vs. LatentLM-L: 2.24)
indicates that constant-norm, directional latents are crucial for high-quality AR decoding.
Overall, SphereAR delivers scale-invariant AR model that sets the best reported FID with far fewer parameters and outperforms diffusion, masked-generation and next-scale baselines. Appendix K shows qualitative results.
4.3 ABLATION STUDY
All variants in this ablation use the same model backbone and training/evaluation configuratio; only the VAE objective/posterior differs. To reduce compute, each VAE is trained on ImageNet for 50 epochs. For the AR stage we use the SphereAR-L. backbone, trained for 50 epochs with a constant learning rate 1 x 1 and batch size 256; all other settings follow Sec. 4.2
S-VAE ys. Diagonal-
The core design of SphereAR is to constrain latents on hypersphere via S-VAE, We compare it with prior VAEs. In particular, we evaluate three settings: (1) Diagonal-
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Table Overall comparison on ImageNet 256x256 class-conditional generation. Abbreviations: AR
next-token (causal) autoregression; Mask. masked generation (next-set); N.S. = next-scale; Diff.
diffusion, An asterisk (*) indicates models trained x84 and evaluated at x56 by resizing.
Model
Type
Order
#Params
#Epochs|
FID
ISt Pre.t Rec.
Discrete Tokens
VQGAN (Esser et al.
2021
AR
raster
1.4B
240
5.20
280.3
ViT-VQGAN
AR
raster
1.7B
240
3.04
227.4
LlamaGen-L Sun al 2024a
AR
raster
343M
300
3.07
256.1
0.83
0.52
LlamaGen-XL: Sun ct 2024g
AR
raster
775M
300
2.62
244.1
0.80
0.57
RandAR-L Pang al 2025
AR
random
343M
300
2.55
288.8
0.81
0.58
RandAR-XI. Pang et al.
2025
AR random
775M
300
2.22
314.2
0.80
0.60
RAR-B (Yu al.
2024
AR hybrid
261M
400
1.95
290.5
0.82
0.58
RAR-L Yu ct al.
2024
AR hybrid
461M
400
1.70
299.5
0.81
0.60
MaskGIT Chang
2022
Mask. random
227M
300
4.02
355.6
0.78
0.50
MAGVIT-vZ Yu et al.
2023
Mask. random
307M
270
1.78
319.4
VAR-d20 Tian et
2024)
N.S.
600M
350
2.57
302.6
0.83
0.56
VAR-d30 Tian et al.
2024
N.S.
2B
350
1.92
323.1
0.82
0.59
Continuous Tokens
LDM-4 (Rombach et
2022
Diff.
400M
3.60
247.7
0.87
0,48
DiT-XL/2 Peebles and Xie
2023
Diff.
675M
400
2.27
278.2
0.83
0.57
SiT-XL/2 Ma et
2024)
Diff.
675M
400
2.06
277.5
0.83
0.59
GIVT (Tschannen
2024
AR
raster
1.67B
500
2.59
0.81
0.57
LatentL.M-L Sun et
2024h
AR
raster
479M
400
253.8 2.24
MAR-B (Li ct al.
2024
Mask. random
208M
800
2.31
281.7
0.82
0.57
MAR-L Li ct al.
2024a
Mask. random
479M
800
1.78
296.0
0.81
0.60
MAR-H
(Li
2024a
Mask. random
943M
800
303.7 1.55
0.81
0.62
SphereAR- (Our)
AR raster
208M
400
1.92
277.8
0.81
0.61
SphereAR-L (Our)
AR
raster
479M
400
1.54
295.9
0.80
0.63
SphereAR- (Our)
AR
raster
943M
400
1.34
300.0
0.80
0.64
Gaussian (A-V)E). A standard diagonal- Gaansssn posterior trained with an up-weighted KL term We sweep four KL weights {0.01, 0.04, 0.08, 0.16), denoted G-01, 6-04, G-08, and G- (2) o-VAE (fixed variance). Following LatentLM (Sun et al.) 2024b), we fix the posterior scale with fixed, non-learned scalar a A(0, C₂), and sample z | (x) + a with ((, I). We sweep Co {0.1, 0.2, 0.5) (denoted F-01, 0-2, F5)).. (3) S- VAE (hyperspherical), A Power Spherical posterior on gd-1 with KL-weight sweep (0.001, 0.004, 0.008), denoted S-01, s 04, S-08.
Fig. 4 (left) plots FID versus CFG for these posteriors. We observe: (1) S- S-VAE is consistently best and most stable across CFG. In particular, S-04 attains the lowest FID and S-08 is close second. (2) Stronger regularization helps diagonal- Gaussian but saturates. Increasing 3 (or C.) improves the curves, yet they become unstable at larger CFG and remain below S-VAE. (3) Fixed variance offers no advantage. V-V variants achieve performance on par with standard diagonal-Gaussin VAEs (e.g., F-02 vs. 04; F-05 vs. G- 08), indicating that fixing the posterior scale does not help.
diasonal-Gaussian
Overall, the above ablation isolates the tokenizer's role: hyperspherical VAEs yield the most robust and best final AR performance.
Post-hoc Normalization on Gaussna As discussed in Sec. 3.1 simple alternative to achieve scale invariance is apply post-hoc normalization latents from Saagonal-Gaussiee posterior. We therefore run an empirical ablation. Starting from G-x models, we project each latent
Most prior VAEs compute the KL by taking a sum over spatial- channel dimensions (h x w x d) and then a batch mean. We instead take mean over all elements (batch and spatial spatial-channel) which lowers the numerical KI. value; to match the effective regularization strength, we therefore use larger KI. weights (our 10-2 roughly matches prior 2 10-9).
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4.5
4.5
FID (lower is better)
4.0
FID (lower is better)
4.0
VAEs
F-01
VAES
3.5
F-02
3.5
G:01
F-05
G-04
G-01
G-08
G-04
G-16
3.0
G-08
3.0
N-01
N-04
S-01
N-08
S-04
N-16
2.5
S-08
2.5
S-04
2.0
2.5
3.0
3.5
4.0
4.5
2.0
2.5
3.0
3.5
4.0
4.5
CFG scale
CFG scale
Figure 4: Impact of VAE variants on generation performance (FID vs. CFG). All variants share the same backbone and training/evaluation setup: only the VAE objective/posterin differs. Left: diagonal- Gaussian with enlarged KL weight (G-01/04/08) and S-VA VAE with Power Spherical posterior (S-01/04/08) Gaussian with post-hoe normalization (N- 0000001/04////1/
a 7-VAE with fixed scale (F-01/02/
Right: additionally includes diagonal-
Table 2: Ablation normalization (applied to the VAE decoder and AR) and posterior family.
No.
Norm. on VAE Decoder Norm. on AR
Posterior
FID4
IST
X
Gaussian
2.97
240.2
Gaussian
2.89
254.3
Gaussian
2.68
257.3
4
Spherical
2.52
258.4
onto the radius-R hypersphere via R Rz/l|z]12, yielding N-01, N-O4, N-088, and N-16. Fig. 4 (right) plots FID versus CPG for these variants.
From these results, we observe: (1) Post-hoc normalization helps. Each N-x improves over its G-x counterpart and more stable higher CFG scales, supporting our motivation that scale- invariant inputs/outputs stabilize AR decoding.
(2) S-VAE remains the best. 5-04 outperforms the best post-hoc-normalized Gaussian (N-C8). This aligns with our analysis: Gaussian with post-hoc normalization optimizes strictly looser variational bound than the hyperspherical ELBO (Appendix B) and induces a non-axially symmetric directional law on gd-1
S-VAE's Component Contributions The above ablations indicate that both the normalization latent tokens and the hyperspherical posterior are important. Because the normalization can affect two interfaces- the VAE decoder's input and the AR model's inputs/outputs- we further isolate their effects by conducting variant that normalizes only the VAE decoder's 's input. As summarized in Table normalization applied to the AR inputs and outputs is more critical. particular, normalizing only the VAE decoder's s input yields modest gain (FID 2.97 2.89, compare No. 1 with No. 2); whereas additionally normalizing AR inputs/outputs produces a larger improvement (FID 2.89 - 2.68, compare No.2 with No. 3). This finding supports our analysis: the AR pathway re-feeds tokens step by step, so scale errors would otherwise accumulate. In contrast, the VAE decoder consumes its input in single pass and does not induce such cascading scale drift. Finally, adopting the Spherical posterior (No. 4) provides a further significant boost (FID 2.68 + 2.52), confirming that aligning the posterior with constant-norm geometry is beneficial.

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5 CONCLUSION
To address variance collapse in continuous-token AR models, we propose SphereAR, whose core idea make all AR inputs and outputs scale-invariant, Concretely, it consists of (1) hyperspherical VAE (S- VAE) that produces latent tokens constrained to a fixed-radius hypersphere; and (2) an autoregressive Transformer with a token-level diffusion head modeling the next-token distribution over hyperspherical latents. During AR training and inference, all inputs and outputs -nnpluuinnnludinn those after CFG rescaling are normalized onto this hypersphere. Our theoretical analysis demonstrates that ccale-invariant inputs and outputs critical to AR modeling. On ImageNet class-conditional generation, SphereAR- (943M) achieves FID 1.34 and SphereAR-I. (479M) achieves FID 1 1.54, surpassing prior diffusion and masked-gent maskeed-eneratton baselines. Ablations point to two key factors: constant-norm AR refeeding and a hyperspherical posterior. With both, S- S-VAE is best, exceeding diagonal-Gaussian o-VAE, and even Etagonal-Gaussae with post-hoc normalization.
Future work While our results substantiate the motivation and design choices of SphereAR, several extensions would further strengthen this work: (i) exploring Riemannian Flow Matching (RFM) (Chen and Lipman (023), which may better align with hyperspherical latent geometry since trajectories of RFM remain on the hypersphere; and (ii) extending SphereAR to more datasets and multimodal applications. We leave these to future work.
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A FIRST-ORDER STABILITY OF RADIUS PROJECTION IN AR
Let NR: IRd (0) - s be the radial projection NR(Z) = Rz/llz|la where $ = {z E Rd |zl|2 = RJ. All lincarizations are taken at reference tokens 2₁ = R ug with || [ūæl12- 1.
R.
Let g denote the pre-normalization next-token map implemented by our model: given the prefix Zck- the causal Transformer produces condition h and the diffusion head returns a provisional latent 2). = g E IRA which is then projected by N- We assume g is continuousl differentiable in a neighborhood of ccck (i.e., C+ its Jacobian 0g/ 0zzck. exists and varies continuously there), which holds for our architecture with smooth activations and fixed-step explicit ODE solvers.
Lemma (Jacobian is the langent-space projector). For 1 R.
2x R²
(N = Pk = Moreover, PE = Pk. P = PR, P& in particular, | 1. Proof. Differentiate NR(z) = Rz |z||2 and evaluate at Zk.
(10)
0, and P&V v for all v E T2, Sy (v: = 0}:
Lemma 2 (First-order scale invariance). For any small Az,
(n Az) 22 Px Az A[|A)]]),
(11)
radial derivatives vanish and tangential derivatives are preserved (eigenvalues and 1). Proof. First-order Taylor expansion with Lemma
Proposition (One-step AR refeeding error, linearized). Let g be the (unnormalized) next-token predictor and define 2s = g(zen), Z = NR((s), and FR = NR Linearizing at Z.c (with Zz Fi (Z<x)) yields
Ck
2k
2o Ps ( JA + m7. mR):
J&
Bacck 2zck
(12)
where eek stacks the prefix errors and my collects local modeling/integration error. Proof. Chain rule with Lemma 2.
Corollary (No scale-channel cascade; norm bound). Writing J4eck 7k ar ZΔ + ty with
ty
2 Sn sd-1
ek Pa(au 2x + tx) tR, and ||ex||2 |PaJall2 ||exll2 + |Pamall
(13)
Thus radial (scale) errors are annihilated before refeeding and cannot cascade along the AR chain; only directional (langential) errors propagate.
Scope.
Statements are local (first-order) at points on gd-1 and assume is C near Zzk- R
B GAUSSIAN POSTERIOR WITH POST-HOC NORMALIZATION: A LOOSER BOUND
We compare the "Gaussian+norm" objective
NR(Z) = Rz 1
Lo(s, vo x) = ( log Po NR(z))] - DKI(10(2) ( p())),
with the spherical ELBO
Ls.vAE(O, v; Egg (u/x)) log Ru) Dri(as(u (ul x) Unif((d-11) 1)),
E gd-1
Write the polar decomposition = (r, u) with = ||z||2 E Exo and u z //|lz|12 E Sd-1 Let (u x) be the pushforward of (z(zz x) under z u. Since NR(Z) Ru depends only on direction, the reconstruction terms coincide:
E us (x x lo
NR(z))] = E (ujx)) log Ru)].
For numerical robustness implement Nz(z) Rz/ max([1z|122 5) with 10-7
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(a) Power Spherical density (axially symmetric about (4).
(b) Gaussian with post-hoc normalization (projected normal ACG): elliptical level sets, not axially symmetric.
Figure 5: Directional distributions on the sphere. Left: Power Spherical respects purely directional geometry- the density depends only on μu u and is axially symmetric, with a single concentration parameter Right: Gaussian norm induces a projected-norma (ACG) law whose level sets follow u EE=uu symmetry axes are determined by E (and the Gaussian mean), so the density is typically elliptical rather than axially symmetric.
For the isotropic Gaussian prior p(z) -O I), one has p(z) p(r) Unif(u) (with p(r) the x-law in IR4). The KL chain rule (disintegration over u) gives
Dz x) I(zz)) = (u x) Unif(s + Egs DKulao(r I u, l[((r)
Combining the two displays yields
CG(0, 1); x) LS.VAE 2/ - (utx)) [DKL(Ro(r u x. [[(())))] s LS.AAE(O, x)xx),
with equality only if (r u, x) = p(r) almost surely (i.e., the posterior's radial law exactly matches the prior and is independent of x, u1. Thus, Gaussian posterior with post-hoc normalization pays an extra nonnegative radial KL penalty while the decoder discards radius; a spherical posterior avoids this mismatch and aligns the bound with the constant-norm constraint.
Remark (axial symmetry vs. projected normal on Sd-1). The Power Spherical (PS) density on the unit sphere has the form frs(u) Za(x) (1 + p uu) with || and x 2 0. axially rotationally symmetric about μr: for any rotation Q with Qμ μa, one has (Qs) = fPs(u). The single scalar к monotonically controls geodesic concentration (with x = 0 yielding the uniform law).
By contrast, Gaussian+norm l|la/l2-inducs projected-normal (Angular Central Gaussian, ACG) directional law. For the zero- mean case (Hg = O), its density is (u) x (u - dd22 level sets follow the quadratic form u - -'-'u and are generally elliptical, not axially symmetric; axial symmetry holds only if E x I (then the law is uniform). With nonzero mean M4 * O. the density also depends on μ Σ u, produc- ing skewed, non-axially symmetric level sets (and, for anisotropic Σ, possible antipodal bimodality when f4 0). Therefore the Power Spherical family matches the intended purely directional ge- ometry on gd-1 whereas Gaussian+norm inherits Euclidean anisotropy from (Mg. Σ) and does not.
z N(H E) in Fd and map to the sphere via
C COMPARISON WITH MAR'S VAE
while keeping the AR backbone, training schedule, and evaluation protocol identical (see Sec. 4.3 As summarized in 3 SAE yields large FID gain (4.54 2.52) and higher IS (241.6 258.4). These results indicate that constant-norm, directional latents from S VAE matcrially strengthen continuous-toles AR generation.
We perform an ablation that swaps our S- V-V for MAR's VAE (Li et al. 2024a)
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swapping our S-VAE for MAR's VAE without retraining (same AR backbone and training/evaluation settings; only the VAE changes).
Table Ablation:
Type
FID
IS1
Pre.t
Rec.
AR + MAR's VAE
4.54
241.6
0.84
0.45
AR + S-VAE (ours)
2.52
258.4
0.82
0.56
Table 4: Comparison of VAE backbones. Training speed measured iterations per second on 8 A100 GPUs, with batch size 256.
Type
#Params
Training Speedt
LPIPS)
FID
IS
Pre.
Rec.
CNN
70M
2.25
0.167
2.63
262.6
0.82
0.55
VIT
170M
7.19
0.178
2.81
251.3
0.82
0.55
Hybrid (ours)
75M
5.81
0.166
2.52
258.4
0.82
0.56
D VAE ARCHITECTURE CNN vs. VIT vs. HYBRID
Most latent VAEs for image generation adopt VQGAN- encoder-decoder (Esser et al. 2021).
ie.. convolutional (CNN) backbone with gownsaupllig//ussapppinn blocks. This design is parameter-effiiient (~77M) but throughput often limited by activation memory and bandwidth on large feature maps, leading to slow training and inference. ViT-VQGAN (Yu al. 2021) replaces the CNN backbone with a Vision Transformer (ViT) to improve efficiency; however, as shown in pure VIT backbone yields weaker generative metrics than a CNN.
Table 4
To balance efficiency and quality, we adopt a hybrid VAE architecture.
The encoder first uses a lightweight CNN stem (with downsampling blocks) for patchification and early spatial mixing, imparting CNN inductive bias while reducing spatial resolution. The resulting patch tokens are then processed by a bidirectional Transformer (ViT blocks) model long-range dependencies. The decoder mirrors this design: a ViT stack refines the latent tokens, followed by a lightweight CNN (with upsampling blocks) for umpatchification and livel-level reconstruction. This hybrid preserves the CNN's 's strong local bias while leveraging the ViT's global receptive field at token resolution, yielding favorable speed-quality tradeoff.
In our implementation, to match the parameter scale of a VQGAN-style CNN encoder-decoder (Fiser ct al., (202T, ), our VAE uses ViT blocks in the encoder and 12 in the decoder, each with hidden size 512. The encoder's CNN stem performs patchification via 4 downsampling stages (overall 16x reduction) with channel widths [64, 64, 128, 256, 512]: the decoder mirrors this with upsampling stages and extra residual block per stage. In total, the S-VAE has 755M parameters.
We compare three encoder-decoder backbones under the same training setup and losses: (i) a VQGAN- style CNN (Esser et al., (02II; (ii) pure ViT (Yu et al.] 2021) (12 Transformer blocks in both encoder and decoder, hidden size 768; ~ ~ 170M params); and (iii) our Hybrid design. To reduce compute, the training follows the setting in Sec. 4.3 We report training throughput (iterations/s) of VAE, perceptual distortion (LPIPS with VGG-16), and downstream ImageNet generative metrics. Results in Table 4 show: VIT is fastest (7.19 it/s) but slightly worse on LPIPS/FID/IS: the CNN is slowest (2.25 it/s) yet competitive in IS; our Hyybrdd retains the best reconstruction (LPIPS 0. 1666 and the best FID (2.52) while running at 5.81 los-about'/ 2.6x faster than the CNN and at 81% of ViT throughput (5.81 1 vs 7.19 it/s). Overall, the hybrid backbone offers the most favorable speed-qualiy trade-off.
E IMAGENET 512 512
Following prior work, we also evaluate ImageNet at 512 x 512 resolution. We train S- S-V at 512 x 512 with hidden dimension d = 16 and KL coefficient 3 = 0.016. For the autoregressive transformer, adopt the SphereAR-L configuration and set the number of cass-conniiional tokens to 64. Unless otherwise noted, all other settings mirror the 256 x 256 experiments. As shown in Table SphereAR-I. achieves an FID of 1.67 competitive with prior models.
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Table 5: Overall comparison on ImageNet 512x512 class-conditional generation.
Model
Type
Order
#Params
#Epochs
FID.
IS
Discrete Tokens
MaskGIT (Caag et al.
2022
Mask. random
227M
300
4.46
342.0
MAGVIT-v2 Yu
2023
Mask. random
307M
270
1.91
324.3
VAR-d36-s (Tan al.
2024
N.S.
2.3B
350
2.62
303.2
Continuous Tokens
DiT-XL/2 Peebles and Xie
2023
Diff. Mask. random
675M
400
3.04
240.8
MAR-L (Li et al. 2024a
481M
800
1.73
279.9
Rp (Our)
AR
raster
481M
400
1.67
292.5
F TRAINING CONVERGENCE EFFICIENCY
2.8
SphereAR-L
MAR-L
2.6
VAR-d20 (350 epochs)
VAR-d20
FID (lower better)
2.4
2.2
SphereAR-L (100 epochs)
2.0
1.8
SphereAR-L (200 epochs)
MAR-L (800 epochs)
SphereAR-L (300 epochs)
1.6
SphereAR- (400 epochs)
1.4
0
100
200
300
400
500
600
700
Training wall-clock time (hours, by 8 NVIDIA A100 GPUs)
Figure 6: Training Convergence Efficiency Comparison. We plot the generative performance (FID, lower is better) function of training cost (wall-clock time) for SphereAR (our model), MAR, and VAR. SphereAR converges to a strong FID score significantly faster than both baselines.
One key advantage of autoregressive (AR) models is their high training efficiency, More specifically, for single training sample, an AR model calculates a loss across all positions of the sequence (i.e., all prediction steps) one forward pass (using teacher forcing). contrast, diffusion or masked-basee models merely compute a loss for single sampled step (e.g., one timestep in diffusion or one mask pattern). Therefore, because diffusion or masked-basse -ased models receive less dense supervisory signal per sample, they often require significantly more epochs to converge.
To explicitly demonstrate this, we additionally evaluated the generative performance of intermediate checkpoints during the training of SphereAR. For fair comparison, all wall-clock training times were measured using 8 NVIDIA A100 GPUs. As shown in Fig. SphereAR converges much faster than MAR and VAR. This efficiency is stark: in terms of epochs. SphereAR (200 epochs) achieves comparable performance to MAR-L (800 epochs). In terms wall-clock time, SphereAR achieves this same performance using only 200 of the training cost required by MAR-I..
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G
INFERENCE SPEED
VAR-d20
SphereAR-L (diff. step- 100)
LilamaGen-XL
SphereAR-L (diff. step=10)
50
MAR-L
SphereAR-L (diff. step=1)
Speed (images/s. higher better)
40
30
20
10
32
64
128
256
Batch size
Figure 7: Inference speed comparison on single A100 GPU. SphereAR's 's speed, which depends on the number of diffusion steps, is benchmarked against MAR (slowest), VAR (fastest), and LlamaGien.
This section benchmarks the inference speed of our model against previous approaches. All bench- marks were performed on a single NVIDIA 100 GPU across various batch sizes. The results, presented in Fig. 7 lead to the following observations: (i) VAR is the fastest model, it only requires dozens of steps to generate an image. The purely autoregressive model LlamaGen is also highly efficient. (ii) MAR is the slowest. This is expected, its architecture requires accessing the full-length token sequence at each sampling step, which is compounded by an additional 100 sampling steps in its diffusion head. (iii) For SphereAR, diffusion head) is significantly faster than MAR but slower than LlamaGen due to the computational overhead of the diffusion head. (iv) We also evaluated two additional SphereAR settings (with 10 and I diffusion steps, respectively), which exhibit progressively faster speeds. This result clearly demonstrates that the primary inference bottleneck lies in the diffusion head.
default configuration (with 100 sampling steps in
These observations highlight several paths for optimization. First, the high sampling cost of the diffusion head can be mitigated by integrating recent few-step diffusion models, such Shorteut- Model (Frans et al. 2024) or MeanFlow (Geng et al. 2025). which dramatically reduce the required number steps. Second, since the core our model a standard autoregressive Transformer (akin to other LLMs), it can directly benefit from established inference optimization libraries (e.g.. vLLM) to further accelerate the token generation process.
H ROBUSTNESS WITH HIGH CFG SCALES
To demonstrate SphereAR's robustness to high CFG scales, we conducted an additional experiment. sampling with progressively higher CFG scales for the best eiggonal-Gaussian
In particular, model (G-16) and our best S- VAE model (S-04) from Sec. 4.3 The results are presented Fig
As shown in the figure, the performance of both models' degrades as the CFG scale increases. However, the rate of degradation for S-04 much slower. This is evidenced by the widening performance gap: a CFG scale of 4.0, the gap between the models is ~1.0 but at CFG scale of 8.0, it widens to 2.0. This result clearly demonstrates that the proposed model is more robust to aggressive CFG scales.
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G-16
8
S-04
FID (lower better)
6
4
3
4.5
5.5
6.5
7.5
8.0
CFG scale
Performance (FID, lower is better) versus CFG scale, comparing S- S-VAE model ("S-04") with the Gaussian baseline ('G-16'). The curves demonstrate that 'S-04' more robust to high CFG values.
Figure 8:
I FAILURE CASES AND LIMITATIONS
We present qualitative analysis of failure modes in Fig. 9 Similar existing methods, SphereAR can produce results with noticeable artifacts, especially in complex classes. However, even in these failure cases, SphereAR-L consistently produces visually superior results with fewer artifacts than the baseline MAR-I.
Despite this relative improvement, both models share common limitations. These primarily include artifacts in human faces and, most prominently, failure to render legible text or other high-frequency details (e.g., street signs, bottle labels), which are often generated as semantically meaningless patterns.
We posit this is largely a data-driven limitation. As noted by MAR (Li et al., 2024a). models trained on academic datasets like ImageNet often show "noticeable gap in visual quality" compared to commercial models trained on massive, diverse web data, especially for categories like coherent text and faces. These failures thus likely attributable to the dataset's limitations rather than fundamental flaw in our architecture.
Due to computational constraints, current evaluation is focused on the ImageNet benchmark. Therefore, assessing the scalability of SphereAR and robustness in more diverse, large-scale data regimes remains critical direction for future investigation.
J USE OF LLMS
We used a large language model (OpenAl's ChatGPT) solely for language Saacahiin---offrradnnng minor rephrasing, and tone consistency. It did not generate influence the scientifie ideas, methods, results, or conclusions. All suggestions were reviewed and edited by the authors, who take full responsibility for the paper's content and integrity.
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TOT
Urpan Wo & GAND non
Dos brustrrus
IGE
SALE OO LENEE
MaTHow
Priee
MUN
(a) MAR-L
(b) SphereAR-L
Figure 9: Failure cases. Similar to existing methods, our model can produce results with noticeable artifacts. We show MAR-L (left) and SphereAR-L (right) results of the same classes.
K MODEL GENERATED EXAMPLES
We show the uncurated 256x256 samples generated by our 479M SphereAR-L, from Fig. 10 to
Fig. 21
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Figure 10: Uncurated 256x256 SphereAR-L samples. Class label: "frilled lizard" (43).
Figure 11: Uncurated 256x256 SphereAR-L samples. Class label: "sulphur-crested cockatoo" (89).
Figure 12: Uncurated 256x256 SphereAR-I. samples. Class label: "golden retriever" (207).
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Figure 13: Uncurated 256x256 SphereAR-L samples. Class label: "Border collie" (232).
Figure 14: Uncurated 256x256 SphereAR-L Class label: "tabby cat" (281).
Figure 15: Uncurated 256x256 SphereAR-I. samples. Class label: "ladybug" (301).
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Figure 16: Uncurated 256x256 SphereAR-L samples. Class label: "chimpanzee" (367).
Figure 17: Uncurated 256x 256 SphereAR-L samples. Class label: "beacon" (437).
Figure 18: Uncurated 256x256 SphereAR-I. samples. Class label: "castle" (483).
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Figure 19: Uncurated 256x256 SphereAR-L samples. Class label: "icecream" (928).
Figure 20:
Uncurated 256x256 SphereAR-L samples. Class label: "cliff" (972).
Figure 21: Uncurated 256x256 sapplee Class label: "coral reef" (973).
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Reviews:
[{"review_id": "TiIquslKX9", "reviewer": "Reviewer_hP4t", "summary": "This paper points out that current autoregressive image generative models based on continuous tokens often performs unsatisfyingly. To tackle this problem, this paper proposes SphereAR. Specifically, the authors analyzes the importance of scale-invariant tokens for AR generation with continuous tokens. Based on this, the authors propose to replace the original VAE with S-VAE, which parameterizes each token on a unit sphere with the same norm. Then the authors combine the S-VAE with an autoregressive transformer and use the diffusion loss to train the model. Experiments show that SphereAR outperforms all mainstream AR and masked generation models with the same model parameter size.", "questions": "1. The author mentions the different behaviors of masked models and autoregressive (AR) models when both use continuous tokens in Sec.3. I am curious about the underlying reason for this difference. In other words, what fundamentally distinguishes masked models from AR models with continuous token?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The motivation is clear, and the analysis regarding the importance of keeping the scale of tokens the same is reasonable.\n2. Using S-VAE is a straightforward yet effective modification.\n3. The experimental results are impressive. The design of the ablation study is also reasonable.", "weaknesses": "Overall, this paper is quite good. I only have the following minor concern.\n1. Since the authors use an AR architecture of transformer, which is different from MAR, it would be better to include the performance of combining this architecture with a standard VAE as a baseline, which is a fair comparison.", "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": "ADHNecpet9", "reviewer": "Reviewer_Li3p", "summary": "This paper presents SphereAR, an autoregressive image generation framework designed to address the instability and performance gap of continuous-token AR models compared to diffusion and masked-generation methods. The key innovation is the use of hyperspherical variational autoencoders (S-VAE) to produce constant-norm latent tokens, enforced throughout AR decoding by projecting all inputs and outputs onto a fixed-radius hypersphere. Theoretical analysis links the scale-invariance to improved stability, and empirical results on ImageNet-1K demonstrate SphereAR’s superior FID and parameter efficiency versus strong baselines. The paper also includes extensive ablations and analyses of architectural choices.", "questions": "Please provide responses to the issues raised in my Weaknesses section.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "1. Theoretical motivation is clear, with the scale-invariance argument spelled out through formal analysis.\n2. Strong Empirical Results:  The clear advantage in both FID and parameter efficiency is validated across multiple regimes (base and large model).\n3. Comprehensive Experimental Methodology: Empirical claims are supported by well-documented implementation details (Sec. 4.1 & Appendix C) and a reproducibility statement promising code and hyperparameter release.", "weaknesses": "1. Limited Discussion of Generalization and Failure Cases: The paper focuses almost exclusively on ImageNet-1K at $256 \\times 256$ resolution (see Section 4 and qualitative figures in Appendix E). There is no analysis of generalization to other domains (e.g., other image datasets, non-class-conditional tasks, or higher resolutions), nor qualitative exploration of failure modes. This raises questions about the robustness and transferability of the claimed benefits.\n2. Empirical Focus Narrowed to FID and Class-Conditional Generation: Table 1 and Figure 4 concentrate on FID, IS, Precision, and Recall. FID is known to have limitations (especially in capturing semantic consistency and diversity)\n3. Ablation Duration and Robustness: The ablation experiments in Section 4.3 are run for a much shorter training regime than full-scale models (“50 epochs instead of 400”, cf. page 8). The methodology may not uncover longer-term stability issues or provide sufficiently representative results for convergence characteristics. Moreover, for Table 2 (\"Component Contributions\"; values not explicitly shown in the main text), further detail and numbers should be presented.", "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": "rO7vM3fkAZ", "reviewer": "Reviewer_xjkB", "summary": "This paper tackles a core limitation in continuous-token autoregressive (AR) image generation—variance collapse caused by heterogeneous latent scales during AR decoding, especially under classifier-free guidance. The authors propose **SphereAR**, which enforces a constant-norm hyperspherical latent space via a hyperspherical VAE (S-VAE) and ensures all AR inputs and outputs are projected back onto a fixed-radius hypersphere. The paper provides theoretical justification showing that normalization eliminates radial error propagation in AR decoding and supports it with extensive empirical evidence on ImageNet-256×256. SphereAR-L (479M) achieves FID 1.54, surpassing diffusion and masked-generation baselines of comparable or even larger parameter scales (e.g., MAR-H, VAR-d30, DiT-XL/2). Ablations further show that hyperspherical latent geometry consistently improves stability and generation quality compared to Gaussian posteriors or post-hoc normalization.", "questions": "Please refer to the \"Weakness\".", "limitations": "", "rating": 6, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "* **Clear motivation with strong theoretical insight.** The paper identifies *scale drift* as the fundamental issue in continuous-token AR and justifies why hyperspherical constraint solves it more elegantly than variance inflation or post-hoc normalization.\n* **Significant empirical improvement.** SphereAR sets a new state-of-the-art among AR models and even outperforms strong diffusion and masked-generation baselines with fewer parameters.\n* **Well-structured ablation study.** Experiments isolate the contribution of (i) spherical posterior vs Gaussian, (ii) normalization applied only at VAE decoder input vs AR loop, and (iii) post-hoc normalization vs intrinsic spherical latent modeling.\n* **Technical contribution beyond engineering tweak.** Introducing hyperspherical VAEs into AR pipelines constitutes a genuinely novel design in continuous-token generative modeling.\n* **Good reproducibility.** Training details, architectural configurations, and evaluation settings are consistently documented.", "weaknesses": "* **Limited analysis beyond ImageNet.** All experiments are class-conditional ImageNet generation at 256×256. It is unclear whether SphereAR scales to higher-resolution tasks or more challenging datasets such as LAION-based text-to-image.\n* **AR decoding speed not fully addressed.** While normalization improves stability, AR inference remains autoregressive with diffusion steps per token, leading to slower sampling compared to parallel masked generation or next-scale diffusion models. A quantitative runtime comparison would be helpful.\n* **Lack of negative results on extreme CFG regimes or resolution scaling.** The theoretical argument suggests strong stability, but empirical evaluation focuses on standard CFG ranges.\n* **S-VAE training complexity.** Hyperspherical posteriors (even using Power Spherical) may increase training difficulty relative to standard diagonal-Gaussian VAEs; the paper acknowledges this but does not quantify the stability or convergence trade-off.", "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": "0B6BVUPNiQ", "reviewer": "Reviewer_LHMS", "summary": "The paper proposes SphereAR, a continuous-token autoregressive image generator that prevents variance collapse by enforcing scale-invariant latents: a hyperspherical VAE constrains each token to a fixed-radius hypersphere, and all AR inputs/outputs—including after classifier-free guidance—are projected back to this sphere. A first-order analysis shows normalization removes radial errors, stabilizing multi-step AR decoding. Coupled with a causal Transformer and a token-level diffusion head, SphereAR sets impressive results on ImageNet 256×256: SphereAR-L achieves FID 1.54, surpassing diffusion and masked-generation baselines and matching MAR-H with ~1/2 the parameters. Ablations confirm hyperspherical latents outperform diagonal-Gaussian, fixed-variance, and Gaussian+post-hoc normalization, and that constant-norm refeeding in AR is the key driver of gains.", "questions": "Please refer to the above section.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "* This paper introduces a simple but powerful principle—enforcing constant-norm latents for continuous-token AR, including after CFG—to remove scale degrees of freedom. It provides a clean theoretical first-order analysis explaining why normalization eliminates radial error accumulation. Besides, it also distinguishes S-VAE from Gaussian+post-hoc normalization and shows why the former is better aligned with the objective.\n\n* Strong empirical evidence on ImageNet 256×256 with clear SOTA FID at comparable or fewer parameters; comprehensive ablations that validate the effectiveness of each proposed component.\n\n* Well-motivated problem statement tied to variance collapse under CFG; intuitive figures and precise method description; appendices with proofs and architectural details; explicit training and inference settings with reproducibility plan.", "weaknesses": "I think this paper does not contain significant weaknesses. I have a few suggestions and questions:\n* It would be better to provide wall-clock (efficiency) comparisons vs MAR/VAR/DiT at matched FID.\n* It would be interesting to see if the proposed method can improve the MAE paradigm as well.", "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)