SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Table 1: Image reconstruction and generation performance on CelebA-HQ  $64^{2}$ . The results of DPF, GEM, and Functa are brought from the corresponding papers.
Model	# Params	Reconstruction	Generation	Inference Efficiency
Learnable	Inference	PSNR ↑	rFID ↓	FID ↓	Precision ↑	Recall ↑	F1 ↑	GFLOPS ↓	fps ↑	Memory (MB) ↓
Functa [13]	3.3 M	2,629.6 K	26.6	28.4	40.4	0.577	0.397	0.470	8.602	332.9	144.1
GEM [12]	99.0 M	921.3 K	-	-	30.4	0.642	0.502	0.563	3.299	559.6	70.3
GASP [14]	34.2 M	83.1 K	-	-	13.5	0.836	0.312	0.454	0.305	1949.3	16.4
DPF [41]	62.4 M	-	-	-	13.2	0.866	0.347	0.495	-	-	-
mNIF (S)	4.6 M	17.2 K	31.5	10.9	21.0	0.787	0.324	0.459	0.069	2958.6	10.2
mNIF (L)	33.4 M	83.3 K	34.5	5.8	13.2	0.902	0.544	0.679	0.340	891.3	24.4
Table 2: Voxel reconstruction and generation performance on ShapeNet  $64^{3}$ . The results of DPF, and GEM are brought from the corresponding papers.
Model	# Params	Reconstruction	Generation	Inference Efficiency
Learnable	Inference	MSE ↓	PSNR ↑	Coverage ↑	MMD ↓	GFLOPS ↓	fps ↑	Memory (MB) ↓
GASP [14]	34.2 M	83.1 K	0.0296	16.5	0.341	0.0021	8.7	180.9	763.1
GEM [12]	99.0 M	921.3 K	0.0153	21.3	0.409	0.0014	207.0	16.7	4010.0
DPF [41]	62.4 M	-	-	-	0.419	0.0016	-	-	-
mNIF(S)	4.6 M	17.2 K	0.0161	20.9	0.430	0.0014	4.4	191.5	642.1
mNIF(L)	46.3 M	83.3 K	0.0166	21.4	0.437	0.0013	21.6	69.6	1513.3
We present quantitative and qualitative results, and also analyze the effectiveness of the proposed approach in comparison to the previous works including Functa [13], GEM [12], GASP [14], and DPF [41]. Note that we adopt the model configurations with the best generation performance for the other baselines.
5.2.1 Quantitative Performance
We compare results from our approach, mNIF, with existing methods in terms of four aspects: reconstruction accuracy, generation quality, model size, and inference efficiency. We present quantitative results from two configurations of our model, mNIF (S) and mNIF (L), which correspond to small and large networks, respectively.
As shown in Tables 1, 2, and 3, our approach consistently achieves state-of-the-art reconstruction and generation quality on the image, voxel, and NeRF scene datasets except the reconstruction on the voxel dataset, in which our model is ranked second. Moreover, mNIF is significantly more efficient than other methods in terms of model size and inference speed in all cases. Considering the model size and inference speed, the overall performance of mNIF is outstanding.
Note that the light configuration of our model denoted as mNIF (S) is most efficient in all benchmarks and also outperforms the baselines in voxel and NeRF. In SRN Cars, mNIF (S) demonstrates huge benefit in terms of efficiency compared to Functa; 199 times less FLOPS, 49 times more fps, and 22 times less memory in single view inference. In the image domain, the performance of mNIF (S) is comparable to GASP in terms of F1 but worse in FID. We conjecture that FID metric overly penalizes blurry images, which is discussed in several works [12, 13, 31, 20].
5.2.2 Qualitative Generation Performance
Figure 2 illustrates generated samples for qualitative evaluation. We visualize the results from our models on each benchmark together with GASP on CelebA-HQ  $64^{2}$  and Functa on SRN Cars to compare results in the image and NeRF scene tasks, respectively. In the image domain, our model, mNIF (L), generates perceptually consistent yet slightly blurry samples compared to ground-truths, while the examples from GASP with a similar FID to ours have more artifacts. In the NeRF scene, rendered views from our model, mNIF (S), and Functa exhibit a similar level of image blur compared to ground-truths. Since both methods use vanilla volumetric rendering, adopting advanced rendering techniques, such as hierarchical sampling [25, 6] and anti-aliasing methods [4, 3] for NeRF scenes, would improve rendering quality.
Table 3: NeRF scene reconstruction and generation performance on SRN Cars. The results of Functa are brought from the corresponding papers.
Model	# Params	Reconstruction	Generation	Inference Efficiency
Learnable	Inference	PSNR ↑	FID ↓	TFLOPS ↓	fps ↑	Memory (GB) ↓
Functa [13]	3.9 M	3.418.6 K	24.2	80.3	1.789	2.0	28.0
mNIF (S)	4.6 M	17.2 K	25.9	79.5	0.009	97.7	1.3
mNIF (L)
mNIF (S)
GASP [14]
Functa [13]
Ground Truth
(a) CelebA-HQ  $64^{2}$
Figure 2: Comparison of generated samples from our models, mNIF (S) and mNIF (L), with the ground-truth and the baseline methods such as GASP [14] and Functa [13], on the CelebA-HQ  $64^{2}$  (2a), ShapeNet  $64^{3}$  (2b) and SRN Cars (2c) datasets.
Ground Truth
(c) SRN Cars
We analyze the characteristics of our trained model for better understanding via latent space exploration and various ablation studies.
5.3.1 Latent Space Interpolation
Figure 3 illustrates the results of interpolation in the latent space; the corner images are samples in the training dataset and the rest of the images are constructed by bilinear interpolations in the latent space. These images demonstrate the smooth transition of sampled data, confirming that the latent vectors learned at the first stage manage to capture the realistic and smooth manifold of each dataset.
5.3.2 Configurations of Mixture Coefficients in mNIF
Table 4(a) examines the performance by varying the types of mixture coefficients: i) shared mixture coefficients across all layers, i.e.,  $\alpha_{m}^{(i)} = \alpha_{m}$ , ii) layer-specific mixture coefficients, and iii) layer-specific mixture coefficients projected from a latent vector as shown in Eq. (10). Among the three mixture settings, the last option yields the best performance and is used for our default setting.
Table 4(b) presents the effect of the number of mixtures, denoted by  $M$ . Reconstruction performance generally improves as the number of mixtures increases, but such a tendency saturates at around  $M = 256$ . We also measure the precision and recall between the ground-truth and reconstructed samples denoted by rPrecision and rRecall, respectively. According to our observation, increasing the size of the mixture improves the recall more than it does the precision.
Table 4(c) shows the effect of the latent dimension on the reconstruction performance of mNIF. Increasing the latent dimension  $H$  leads to improving the reconstruction performance in both the train and test splits due to the degree-of-freedom issue.
(a) CelebA-HQ  $64^{2}$
Figure 3: Reconstructed images via latent vector interpolations. The examples at the four corners surrounded by green boxes are reconstructed instances using the latent vectors on CelebA-HQ  $64^{2}$ , ShapeNet  $64^{3}$ , and SRN Cars. The rest of samples are generated via bilinear interpolations of latent vectors corresponding to the images at the corners.
(b) ShapeNet  $64^{3}$
(c) SRN Car
Table 4: Ablation study on the mixture coefficient configuration of mNIF on CelebA-HQ  $64^{2}$ . Note that  $L$ ,  $W$ ,  $M$ , and  $H$  denote hidden layer depth, hidden layer width, the number of mixtures, and the dimensionality of a latent coefficient vector, respectively.
Exp	Mixture	(L, W, M, H)	# Params	Train	Test
Learnable	Inference	PSNR ↑	rFID ↓	rPrecision ↓	rRecall ↓	PSNR ↑	rFID ↓
(a)	Shared	(2,64,64,64)	557.2 K	8.7 K	22.20	50.70	0.497	0.003	22.11	55.87
Layer-specific	(2,64,64,256)	557.2 K	8.7 K	24.45	38.23	0.461	0.013	24.35	42.65
Latent	(2,64,64,256)	623.0 K	8.7 K	25.27	31.67	0.534	0.040	25.09	36.85
(b)	Latent	(2,64,16,256)	155.8 K	22.02	53.01	0.433	0.001	21.84	57.23
(2,64,64,256)	623.0 K	25.27	31.67	0.534	0.040	25.09	36.85
(2,64,256,256)	2.5 M	26.64	24.62	0.640	0.134	25.74	32.17
(2,64,1024,256)	10.0 M	26.84	23.71	0.642	0.155	25.85	32.14
(c)	Latent	(5,128,256,256)	21.8 M	31.17	10.65	0.890	0.750	25.35	31.58
(5,128,256,512)	22.3 M	83.3 K	32.11	8.96	0.918	0.845	27.92	24.79
(5,128,256,1024)	23.2 M	32.71	8.09	0.935	0.893	29.45	22.05
5.3.3 Diversity in Learned Neural Bases
To investigate the diversity of the basis functions per layer, Figure 4 visualizes the absolute value of the pairwise cosine similarity between the weight matrices of the basis models. In the cosine similarity matrices, an element with a small value indicates that the corresponding two neural basis functions are nearly orthogonal to each other, which serves as evidence of diversity in the learned basis functions. The visualization also reveals the following two characteristics. First, the lower layers are more correlated than the upper ones partly because low-level features inherently have less diversity. Second, a small subset of neural bases have high correlations with others. We hypothesize that the concentration of high correlation among the small subset of neural bases could be mitigated by introducing a loss function enforcing orthogonality.
5.3.4 Context Adaptation Strategies
We perform the ablation study with respect to diverse optimization strategies and demonstrate the resulting reconstruction performance in Table 5. We train small mNIFs with  $(L, W, M, H) = (2, 64, 256, 256)$  on the CelebA-HQ  $64^2$  dataset. It is clear that the meta-learning approach using second-order gradient computation yields favorable results compared to other options. Running more inner-loop iterations marginally improves performance, but at the cost of additional computational complexity in time and space. Therefore, we set the number of inner-loop iterations to 3 for all meta-learning experiments.
Interestingly, the auto-decoding strategy surpasses meta-learning with first-order gradient computation. Note that auto-decoding is more efficient than other methods in terms of both speed and memory usage when evaluated under the same optimization environment. This efficiency of auto-decoding is
Figure 4: Visualization of the absolute value of cosine similarity between neural basis functions, observed in five different layers. We compute the similarity of the hidden layers in mNIF (L) trained on CelebA-HQ  $64^{2}$ . Diagonal values in these matrices are set to zero for visualization.
Table 5: Study on context adaptation strategy.
Strategy	Second-order	\(N_{inner}\)	PSNR ↑	rFID ↓
Auto-decoding	-	-	24.68	34.31
\(\widetilde{x}\)	3	22.68	57.27
Meta-learning	✓	3	26.64	24.64
✓	5	26.77	24.37
Table 6: Study on longer training of mNIF.
Epoch	Reconstruction	Generation
PSNR ↑	rFID ↓	FID ↓
400	32.11	8.96	14.96
800	33.10	7.31	14.30
1200	33.61	6.91	15.25
particularly beneficial for NeRF scene modeling, which requires a large number of input coordinate queries. Our model runs on a single GPU for training on 8 NeRF scenes per batch while Functa needs 8 GPUs for the same purpose.
5.3.5 Longer Training of Context Adaptation
Table 6 shows the performance of our approach, mNIF, with respect to the number of epochs for the first stage training under the identical stage 2 training environment. When we train mNIF (L) with  $(L,W,M,H) = (5,128,256,512)$  on the CelebA-HQ  $64^{2}$  dataset, reconstruction performance is positively correlated with training length but longer training may be prone to overfitting considering the trend of generalization performance.
We presented a novel approach for generative neural fields based on a mixture of neural implicit functions (mNIF). The proposed modulation scheme allows us to easily increase the model capacity by learning more neural basis functions. Despite the additional basis functions, mNIF keeps the network size for inference compact by taking advantage of a simple weighted model averaging, leading to better optimization in terms of latency and memory footprint. The trained model can sample unseen data in a specific target task, where we adopt the denoising diffusion probabilistic model [18, 13] for sampling context vectors  $\phi$ . We have shown that our method achieves competitive reconstruction and generation performance on various domains including image, voxel data, and NeRF scene with superior inference speed.
The proposed method currently exhibits limited scalability beyond fine-grained datasets. We hypothesize that the latent space in mNIF trained on a dataset with diverse examples, such as CIFAR-10, is not sufficiently smooth compared to other datasets to be generalized by adopting a diffusion process. A possible solution to these challenges is to incorporate local information into the architecture based on neural implicit representations, as proposed in recent research [39, 5]. We reserve this problem for future work.
This work was partly supported by the National Research Foundation of Korea (NRF) grant [No. 2022R1A2C3012210, No.2022R1A5A708390811] and the Institute of Information & communications Technology Planning & Evaluation (IITP) grant [NO.2021-0-0268, No. 2021-0-01343], funded by the Korea government (MSIT).
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A.1 Benchmark
A.1.1 Images
We use CelebA-HQ dataset [21] for our work. We divide entire images into 27,000 images for train and 3,000 images for test split which is provided by Functa [13]. We use the pre-process dataset from the link  ${}^{1}$  . To measure the quality of generated images, we compute Fréchet inception distance (FID) score [17], precision and recall [32, 27] between sampled images and images in a train split.
A.1.2 Voxels
We utilize the ShapeNet dataset [8] for 3D voxel generation from IM-Net [10]. It has 35,019 samples in the train split and 8,762 samples in the test split. The dataset contains a voxel with  $64^3$  and 16,384 points extracted near the surface. To evaluate the generative model, we sample 8,764 shapes, which is the size of the test set, and generate all points in the voxel. Then, we compute 2,048 dimension mesh features from the voxel following the protocol in [10]. Generation performance with coverage and maximum mean discrepancy (MMD) metrics [1] is evaluated using Chamfer distance on test split. We follow the evaluation procotol from generative manifold learning (GEM) [12].
A.1.3 Neural Radiance Field (NeRF)
We adopt SRN Cars dataset [34] for NeRF modeling. SRN cars dataset has a train, validation, and test split. Train split has 2,458 scenes with  $128 \times 128$  resolution images from 50 random views. Test split has 704 scenes with  $128 \times 128$  images from 251 fixed views in the upper hemisphere.
We adopt the evaluation setting of NeRF scene from functa which utilizes 251 fixed views from the test split. Consequently, we can compute FID score based on rendered images and images in a test split with equivalent view statistics.
A.2 Implicit Neural Representation with Mixture of Neural Implicit Functions
A.2.1 Images
We train our mixtures of Neural Implicit Functions (mNIF) on images from train split of CelebA-HQ  $64^2$  dataset. Generative implicit neural representation (INR) for image takes 2D input coordinates  $(x_{1},x_{2})$  and returns 3D RGB values  $(y_{R},y_{G},y_{B})$ . During optimization, we use dense sampling, which queries all input coordinates  $(4096 = 64\times 64)$  to the network inputs, for image benchmark. We use mNIF configuration with the number of hidden layers  $L = 5$ , the channel dimension of hidden layers  $W = 128$ , the number of mixtures  $M = 384$  and the latent vector dimension  $H = 512$ . throughout our experiments. We use meta learning for training the mNIF on CelebA-HQ  $64^2$ . We use Adam [22] optimizer with the outer learning rate  $\epsilon_{\mathrm{outer}} = 1.0\times 10^{-4}$  and batch size 32. We use a cosine annealing learning rate schedule without a warm-up learning rate. We take the best mNIF after optimizing the model over 800 epochs with inner learning rates  $\epsilon_{\mathrm{inner}}\in \{10.0,1.0,0.1\}$ . The configuration of mNIF (S) is  $(L,W,M,H) = (4,64,256,1024)$  and mNIF (L) (5,128,384,512). The best performed model uses  $\epsilon_{\mathrm{inner}} = 1.0$ .
A.2.2 Voxels
We train our model on ShapeNet  $64^{3}$  dataset. Our generative neural field for voxel takes 3D input coordinates  $(x_{1},x_{2},x_{3})$  and returns  $y_{\sigma}$  indicating whether queried coordinate are inside or outside. For model optimization, we use sub-sampling with 4,096 points in a voxel with  $64^{3}$  from the points. Then, we use all 16,384 points for constructing latent vectors. Note that despite relying on sub-sampling, the 16,384 points sampled from the near surface of a voxel are dense enough to effectively represent its content within the voxel. We use meta learning for training the mNIF on ShapeNet  $64^{3}$ . We use mNIF (S) and (L) configuration with  $(L,W,M,H) = (4,64,256,512)$  and  $(L,W,M,H) = (5,128,512,1024)$ . We take the best mNIF after optimizing the model over 400 epochs with inner learning rates  $\epsilon_{\mathrm{inner}} \in \{10.0,1.0,0.1\}$ .
<https://drive.google.com/drive/folders/11Vz0fqHS2rXDb5pprgTjpD7S2BAJhi1P
A.2.3 NeRF Scenes
We train our model on SRN Cars dataset. We use a simplified NeRF protocol without a view dependency on neural fields. So, generative neural field returns 4D outputs, RGB values and density  $(y_{R}, y_{G}, y_{B}, y_{\sigma})$ , with given 3D input coordinates  $(x_{1}, x_{2}, x_{3})$ . We use no activation function for RGB output and exponential linear unit (ELU) [11] for density output as suggested in Functa. During optimization, we use sparse sampling with 32 views per scene, 512 pixels per view and 32 points per ray for each scene. We use auto-decoding procedure for efficient optimization instead of meta-learning procedure to avoid the cost of second-order gradient computation. In auto-decoding procedure, we simply harvest latent vectors for the dataset jointly trained during the optimization procedure. We use mNIF (S) configuration with  $(L, W, M, H) = (4, 64, 256, 128)$ . We use Adam [22] optimizer with outer learning rate  $\epsilon_{\mathrm{outer}} = 1.0 \times 10^{-4}$  and batch size 8. We use cosine annealing without a warm-up learning rate for the learning rate schedule. We use inner learning rate  $\epsilon_{\mathrm{inner}} = 1.0 \times 10^{-4}$  and use weight decay on latent vector  $\phi_{j}$  to regularize the latent space as in auto-decoding framework. We take the best mNIF after optimizing the model over 2000 epochs.
A.2.4 Weight Initialization of mNIF
Weight initialization is crucial for optimizing INR, such as SIREN [33]. We enforce mixture coefficients at the start as the inverse of the number of mixtures  $\alpha_{m}^{(i)} = 1 / M$  to give the equivalent importance for all implicit bases in mixtures. For latent mixture coefficients, we set bias of projection matrix into  $1 / M$  to satisfy above condition.
A.3 Denoising Diffusion Process on Latent Space
We train unconditional generation models for latent space for image, voxel and NeRF scene for generating latent coefficients vector. We use the equivalent setting for the training diffusion process throughout all experiments and benchmarks. We implement DDPM [18] on latent vector space based on residual MLP [13] from Functa as a denoising model. The channel dimension of residual MLP is 4096 and the number of block is 4. We use Adam optimizer and set learning rate  $1.0 \times 10^{-4}$  and batch size 32. We train model 1000 epochs on each dataset and schedule the learning rate with cosine annealing without a warm-up learning rate. We use  $T = 1000$  time steps for diffusion process.
A.4 Libraries and Code Repository
Our implementation is based on PyTorch  $1.9^{2}$  with CUDA 11.3 and PyTorch Lightning 1.5.7. Our major routines are based on the code repositories for Functa, SIREN, latent diffusion model, guided diffusion and HQ-Transformer. For inference efficiency, we count FLOPS with the fvcore library and check inference time and memory consumption on GPU by PyTorch Profiler. The efficiency of mNIF are evaluated on NVIDIA Quadro RTX 8000.
B Analysis
B.1 Visualization
We provide additional generation examples in Figures 5 and 6 and additional interpolation results of the proposed algorithm in Figures 7 and 8.
PyTorch (https://pytorch.org)
3PyTorch Lightning (https://www.pytorchlightning.ai)
$^{4}$  Functa (https://github.com/deepmind/functa)
5IREN (https://github.com/vsitzmann/siren)
6Latent Diffusion Model (https://github.com/CompVis/latent-diffusion)
Guided diffusion (https://github.com/openai/guided-diffusion)
$^{8}$ HQ-Transformer (https://github.com/kakaobrain/hqtransformer)
${}^{9}$  fvcore (https://github.com/facebookresearch/fvcore/blob/main/docs/flop_count.md)
$^{10}$ PyTorch Profiler (https://pytorch.org/tutorials/recipes/recipes/profilerrecipe.html)
As discussed in conclusion, we observe that our model has limited scalability beyond fine-grain dataset; our model works well on reconstruction while not on generation on CIFAR10. To analyze the generation on CIFAR10, we introduce an alternative sampling scheme to replace the latent diffusion model described in our main paper. Drawing inspiration from the sampling strategy in generative manifold learning (GEM), we employed latent space interpolation between two samples: one random sample from the training set and another random sample from its neighborhood. The resulting context vector, is formed by a simple interpolation:
$$
\phi_ {\text {s a m p l e}} = \alpha \cdot \phi_ {i} + (1 - \alpha) \cdot \phi_ {\mathcal {N} (i)}, \tag {11}
$$