CACFG: Curvature-Aware Classifier-Free Guidance and Optimal Control

Max Collins; Dasith de Silva Edirimuni; Jordan Vice; Tim French; Ajmal Mian

Figure 4: Tweedie’s magnitude (

\lVert\hat{\boldsymbol{x}}_{0}(t,\boldsymbol{x}_{t},\xi)\rVert_{2}

) of SD1.5 CFG and CACFG (ours) sampling trajectories at guidance strengths

\omega\in\{5,7.5,10,15\}

on (a) ImageNet ( Russakovsky et al., 2015 ) and (b) COCO ( Lin et al., 2014 ) . Red paths violate the theoretical

95\%

upper magnitude bound

r_{u}=129.16

(black dashed line); the violating fraction is annotated in each panel. The proportion of violating paths grows sharply with guidance strength for CFG, and this growth is suppressed by our curvature-aware correction, with its time-varying upper bound (red dashed line). We recommend zooming in for easier viewing.

Figure 5: Empirical distribution of SD1.5 VAE encodings (

\lVert\boldsymbol{x}_{0}\rVert_{2}

) of the ImageNet validation set, i.e. the distribution of

\lVert\boldsymbol{x}_{0}\rVert_{2}

. Red line denotes the theoretical chi density implied by the Gaussian hypothesis in Section 5 .

Strength	Guidance	FID $\downarrow$	NIQE $\downarrow$	BRISQUE $\downarrow$	TReS $\uparrow$	CLIP $\uparrow$	HPSv2 $\uparrow$	Density $\uparrow$	Coverage $\uparrow$	Saturation $\downarrow$
5	CFG	${\color[rgb]{0,0.88,0}\mathbf{13.67}}$	$4.06$	$14.98$	$83.24$	$24.851$	$25.076$	${\color[rgb]{0,0.88,0}\mathbf{1.128}}$	$0.453$	$0.299$
	CACFG	$13.70$	${\color[rgb]{0,0.88,0}\mathbf{4.05}}$	${\color[rgb]{0,0.88,0}\mathbf{14.83}}$	${\color[rgb]{0,0.88,0}\mathbf{83.47}}$	$24.849$	$25.070$	$1.124$	${\color[rgb]{0,0.88,0}\mathbf{0.454}}$	${\color[rgb]{0,0.88,0}\mathbf{0.295}}$
	CACFG*	${\color[rgb]{0,0.88,0}\mathbf{13.67}}$	${\color[rgb]{0,0.88,0}\mathbf{4.05}}$	$14.98$	$83.27$	${\color[rgb]{0,0.88,0}\mathbf{24.856}}$	${\color[rgb]{0,0.88,0}\mathbf{25.081}}$	$1.123$	${\color[rgb]{0,0.88,0}\mathbf{0.454}}$	$0.299$
7.5	CFG	$15.12$	$4.27$	$15.47$	$82.80$	${\color[rgb]{0,0.88,0}\mathbf{25.251}}$	$25.680$	$1.139$	$0.443$	$0.338$
	CACFG	${\color[rgb]{0,0.88,0}\mathbf{14.97}}$	${\color[rgb]{0,0.88,0}\mathbf{4.22}}$	${\color[rgb]{0,0.88,0}\mathbf{14.59}}$	${\color[rgb]{0,0.88,0}\mathbf{83.80}}$	$25.243$	$25.653$	${\color[rgb]{0,0.88,0}\mathbf{1.140}}$	${\color[rgb]{0,0.88,0}\mathbf{0.445}}$	${\color[rgb]{0,0.88,0}\mathbf{0.316}}$
	CACFG*	$15.11$	$4.27$	$15.33$	$82.85$	$25.249$	${\color[rgb]{0,0.88,0}\mathbf{25.687}}$	$1.132$	$0.444$	$0.337$

Appendix

Table 1: Image quality for SD1.5 samples generated for ImageNet using CFG ( Ho & Salimans, 2022 ) and CACFG with varying upper bound limits,

r_{u}

. CACFG denotes the

r_{u}=129.16

bound used in the main paper, while CACFG* is the

r_{u}=211.19

bound derived from the maximum SD1.5 VAE encoding of the ImageNet validation dataset. The best result for each dataset and guidance strength is bold .

$\boldsymbol{\gamma}$	FID $\downarrow$	NIQE $\downarrow$	BRISQUE $\downarrow$	TReS $\uparrow$	CLIP $\uparrow$	HPSv2 $\uparrow$	Density $\uparrow$	Coverage $\uparrow$	Saturation $\downarrow$
$0.0$	$16.95$	$5.44$	$23.59$	$85.15$	$24.835$	$23.844$	$0.938$	$0.395$	$0.194$
$0.1$	$14.53$	$4.56$	$17.36$	$86.28$	$25.159$	$25.069$	$1.111$	$0.444$	$0.236$
$0.2$	$14.54$	$4.32$	$15.28$	$85.65$	$25.199$	$25.403$	$1.142$	$0.446$	$0.266$
$0.3$	$14.72$	$4.24$	$14.64$	$84.96$	$25.233$	$25.550$	$1.140$	$0.448$	$0.289$
$0.4$	$14.89$	$4.22$	$14.50$	$84.29$	$25.242$	$25.621$	$1.140$	$0.447$	$0.305$
$0.5$	$14.97$	$4.22$	$14.59$	$83.80$	$25.243$	$25.653$	$1.140$	$0.445$	$0.316$

Appendix

Table 2: Image quality for SD1.5 ( Rombach et al., 2022 ) samples generated for ImageNet ( Russakovsky et al., 2015 ) dataset using CACFG across varying curvature-aware blending parameter values.

$\boldsymbol{\gamma}$	FID $\downarrow$	NIQE $\downarrow$	BRISQUE $\downarrow$	TReS $\uparrow$	CLIP $\uparrow$	HPSv2 $\uparrow$	Density $\uparrow$	Coverage $\uparrow$	Saturation $\downarrow$
$0.0$	$35.84$	$4.96$	$26.93$	$78.06$	$24.751$	$23.531$	$0.863$	$0.143$	$0.139$
$0.1$	$29.85$	$4.48$	$26.04$	$81.82$	$25.213$	$25.409$	$1.073$	$0.174$	$0.165$
$0.2$	$29.22$	$4.42$	$26.05$	$83.37$	$25.274$	$25.924$	$1.133$	$0.183$	$0.182$
$0.3$	$29.09$	$4.43$	$26.21$	$83.96$	$25.291$	$26.114$	$1.147$	$0.184$	$0.193$
$0.4$	$29.02$	$4.44$	$26.45$	$84.22$	$25.322$	$26.234$	$1.117$	$0.181$	$0.201$
$0.5$	$29.02$	$4.47$	$26.53$	$84.31$	$25.361$	$26.294$	$1.140$	$0.184$	$0.206$

Appendix

Table 3: Image quality for SDXL ( Podell et al., 2024 ) samples generated for ImageNet ( Russakovsky et al., 2015 ) dataset using CACFG across varying curvature-aware blending parameter values.

Figure 6: Performance of CFG, CFG++, and IntCFG with (solid line) and without (dotted line) our curvature-aware correction using SD1.5 on (a) ImageNet and (b) COCO. Some metric axes have been flipped so that the improvement direction is vertically up the page. Note at larger guidance strengths FID, NIQE, BRISQUE, TReS, Density, Coverage, and CLIP are all improved by our curvature-aware correction. We recommend zooming in for easier viewing.

Figure 7: Performance of CFG, CFG++, and IntCFG with (solid line) and without (dotted line) our curvature-aware correction using SDXL on (a) ImageNet and (b) COCO. Some metric axes have been flipped so that the improvement direction is vertically up the page. Note at larger guidance strengths FID, NIQE, BRISQUE, TReS, Density, and Coverage are all improved by our curvature-aware correction. We recommend zooming in for easier viewing.

Figure 8: CFG ( Ho & Salimans, 2022 ) , CFG++ ( Chung et al., 2025 ) , and IntCFG ( Kynkäänniemi et al., 2024 ) visualised against their curvature-aware variants (ours): CACFG, CACFG++, IntCACFG. Samples are generated using SD1.5 ( Rombach et al., 2022 ) targeting the ImageNet ( Russakovsky et al., 2015 ) dataset with a guidance strength of

\omega=15

.

Figure 9: CFG ( Ho & Salimans, 2022 ) and CACFG (ours) samples generated using SD1.5 ( Rombach et al., 2022 ) targeting the ImageNet ( Russakovsky et al., 2015 ) dataset. Guidance scale varies across

\omega=5,7.5,10,15

.

Figure 10: CFG ( Ho & Salimans, 2022 ) , CFG++ ( Chung et al., 2025 ) , and IntCFG ( Kynkäänniemi et al., 2024 ) visualised against their curvature-aware variants (ours): CACFG, CACFG++, IntCACFG. Samples are generated using SD1.5 ( Rombach et al., 2022 ) targeting the COCO ( Lin et al., 2014 ) dataset with a guidance strength of

\omega=15

.

Figure 11: CFG ( Ho & Salimans, 2022 ) and CACFG (ours) samples generated using SD1.5 ( Rombach et al., 2022 ) targeting the COCO ( Lin et al., 2014 ) dataset. Guidance scale varies across

\omega=5,7.5,10,15

.

Figure 12: CFG ( Ho & Salimans, 2022 ) , CFG++ ( Chung et al., 2025 ) , and IntCFG ( Kynkäänniemi et al., 2024 ) visualised against their curvature-aware variants (ours): CACFG, CACFG++, IntCACFG. Samples are generated using SDXL ( Podell et al., 2024 ) targeting the ImageNet ( Russakovsky et al., 2015 ) dataset with a guidance strength of

\omega=25

.

Figure 13: CFG ( Ho & Salimans, 2022 ) and CACFG (ours) samples generated using SDXL ( Podell et al., 2024 ) targeting the ImageNet ( Russakovsky et al., 2015 ) dataset. Guidance scale varies across

\omega=10,15,20,25

.

Figure 14: CFG ( Ho & Salimans, 2022 ) , CFG++ ( Chung et al., 2025 ) , and IntCFG ( Kynkäänniemi et al., 2024 ) visualised against their curvature-aware variants (ours): CACFG, CACFG++, IntCACFG. Samples are generated using SDXL ( Podell et al., 2024 ) targeting the COCO ( Lin et al., 2014 ) dataset with a guidance strength of

\omega=25

.

Figure 15: CFG ( Ho & Salimans, 2022 ) and CACFG (ours) samples generated using SDXL ( Podell et al., 2024 ) targeting the COCO ( Lin et al., 2014 ) dataset. Guidance scale varies across

\omega=10,15,20,25

.

Dataset	Strength	Guidance	FID $\downarrow$	NIQE $\downarrow$	BRISQUE $\downarrow$	TReS $\uparrow$	CLIP $\uparrow$	HPSv2 $\uparrow$	Density $\uparrow$	Coverage $\uparrow$	Saturation $\downarrow$
ImageNet	5	CFG	${\color[rgb]{0,0.88,0}\mathbf{13.67}}$	$4.06$	$14.98$	$83.24$	${\color[rgb]{0,1,0}24.851}$	${\color[rgb]{0,1,0}25.076}$	${\color[rgb]{0,0.88,0}\mathbf{1.128}}$	$0.453$	$0.299$
		CFG++	${\color[rgb]{0,1,0}13.88}$	$4.05$	$15.27$	$83.31$	$24.842$	${\color[rgb]{0,0.88,0}\mathbf{25.096}}$	${\color[rgb]{0,1,0}1.093}$	$0.447$	$0.291$
		IntCFG	${\color[rgb]{0,1,0}13.86}$	$4.07$	$15.00$	$83.04$	$24.859$	${\color[rgb]{0,1,0}25.079}$	$1.112$	${\color[rgb]{0,1,0}0.449}$	$0.295$
		CACFG	$13.70$	${\color[rgb]{0,1,0}4.05}$	${\color[rgb]{0,0.88,0}\mathbf{14.83}}$	${\color[rgb]{0,0.88,0}\mathbf{83.47}}$	$24.849$	$25.070$	$1.124$	${\color[rgb]{0,0.88,0}\mathbf{0.454}}$	${\color[rgb]{0,1,0}0.295}$
		CACFG++	$13.90$	${\color[rgb]{0,0.88,0}\mathbf{4.04}}$	${\color[rgb]{0,1,0}15.19}$	${\color[rgb]{0,1,0}83.44}$	${\color[rgb]{0,1,0}24.843}$	$25.090$	$1.090$	${\color[rgb]{0,1,0}0.449}$	${\color[rgb]{0,0.88,0}\mathbf{0.289}}$
		IntCACFG	${\color[rgb]{0,1,0}13.86}$	${\color[rgb]{0,1,0}4.06}$	${\color[rgb]{0,1,0}14.87}$	${\color[rgb]{0,1,0}83.29}$	${\color[rgb]{0,0.88,0}\mathbf{24.862}}$	$25.072$	${\color[rgb]{0,1,0}1.113}$	${\color[rgb]{0,1,0}0.449}$	${\color[rgb]{0,1,0}0.290}$

Appendix

Table 4: Image quality for SD1.5 samples generated for ImageNet and COCO datasets across a range of guidance strengths and schedules. Best result for each dataset and guidance strength is bold , and the best value between curvature-aware and non-curvature-aware guidance is given in light green , i.e., colour denotes whether CA or non-CA methods were favoured.

Dataset	Strength	Guidance	FID $\downarrow$	NIQE $\downarrow$	BRISQUE $\downarrow$	TReS $\uparrow$	CLIP $\uparrow$	HPSv2 $\uparrow$	Density $\uparrow$	Coverage $\uparrow$	Saturation $\downarrow$
ImageNet	10	CFG	$29.60$	$4.66$	$27.42$	${\color[rgb]{0,1,0}85.59}$	$25.548$	${\color[rgb]{0,0.88,0}\mathbf{26.815}}$	${\color[rgb]{0,0.88,0}\mathbf{1.137}}$	${\color[rgb]{0,0.88,0}\mathbf{0.182}}$	$0.240$
		CFG++	$29.74$	$4.63$	$27.21$	${\color[rgb]{0,0.88,0}\mathbf{86.03}}$	${\color[rgb]{0,1,0}25.508}$	${\color[rgb]{0,1,0}26.762}$	$1.071$	${\color[rgb]{0,1,0}0.174}$	$0.231$
		IntCFG	$30.12$	$4.67$	$27.55$	$85.91$	${\color[rgb]{0,1,0}25.508}$	${\color[rgb]{0,1,0}26.722}$	$1.074$	$0.172$	$0.234$
		CACFG	${\color[rgb]{0,0.88,0}\mathbf{29.47}}$	${\color[rgb]{0,1,0}4.64}$	${\color[rgb]{0,1,0}27.35}$	$85.57$	${\color[rgb]{0,0.88,0}\mathbf{25.563}}$	$26.789$	$1.102$	$0.177$	${\color[rgb]{0,1,0}0.229}$
		CACFG++	${\color[rgb]{0,1,0}29.55}$	${\color[rgb]{0,0.88,0}\mathbf{4.59}}$	${\color[rgb]{0,0.88,0}\mathbf{27.04}}$	$86.01$	$25.491$	$26.754$	${\color[rgb]{0,1,0}1.074}$	${\color[rgb]{0,1,0}0.174}$	${\color[rgb]{0,1,0}0.227}$
		IntCACFG	${\color[rgb]{0,1,0}29.91}$	${\color[rgb]{0,1,0}4.64}$	${\color[rgb]{0,1,0}27.33}$	${\color[rgb]{0,1,0}86.01}$	$25.488$	$26.675$	${\color[rgb]{0,1,0}1.078}$	${\color[rgb]{0,1,0}0.174}$	${\color[rgb]{0,0.88,0}\mathbf{0.223}}$

Appendix

Table 5: Image quality for SDXL samples generated for ImageNet and COCO datasets across a range of guidance strengths and schedules. Best result for each dataset and guidance strength is bold , and the best value between curvature-aware and non-curvature-aware guidance is given in light green , i.e., colour denotes whether CA or non-CA methods were favoured.

Dataset	Guidance	Strength	FID $\downarrow$	NIQE $\downarrow$	BRISQUE $\downarrow$	TReS $\uparrow$	CLIP $\uparrow$	HPSv2 $\uparrow$	Density $\uparrow$	Coverage $\uparrow$	Saturation $\downarrow$
ImageNet	CFG	5	${\color[rgb]{0,0.88,0}\mathbf{13.67}}$	${\color[rgb]{0,1,0}4.06}$	$14.98$	$83.24$	$24.851$	$25.076$	$1.128$	${\color[rgb]{0,1,0}0.453}$	$0.299$
		7.5	$15.12$	$4.27$	$15.47$	$82.80$	$25.251$	$25.680$	${\color[rgb]{0,1,0}1.139}$	$0.443$	$0.338$
		10	$16.22$	$4.44$	$17.13$	$81.41$	$25.391$	$25.967$	$1.084$	$0.424$	$0.371$
		15	$17.78$	$4.75$	$21.95$	$78.14$	${\color[rgb]{0,1,0}25.491}$	${\color[rgb]{0,0.88,0}\mathbf{26.214}}$	$0.994$	$0.391$	$0.424$
	APG	5	$14.07$	$4.19$	$17.07$	$84.85$	$24.240$	$23.826$	$1.009$	$0.427$	${\color[rgb]{0,0.88,0}\mathbf{0.230}}$
		7.5	$14.06$	$4.51$	$17.04$	$85.55$	$24.806$	$24.567$	$1.091$	$0.443$	${\color[rgb]{0,1,0}0.233}$

Appendix

Table 6: Image quality for SD1.5 ( Rombach et al., 2022 ) samples generated for ImageNet ( Russakovsky et al., 2015 ) and COCO ( Lin et al., 2014 ) datasets using CFG ( Ho & Salimans, 2022 ) , CACFG, and APG ( Sadat et al., 2025 ) . Best result for each dataset are given in bold green and second best in light green .

Dataset	Guidance	Strength	FID $\downarrow$	NIQE $\downarrow$	BRISQUE $\downarrow$	TReS $\uparrow$	CLIP $\uparrow$	HPSv2 $\uparrow$	Density $\uparrow$	Coverage $\uparrow$	Saturation $\downarrow$
ImageNet	CFG	10	$29.60$	$4.66$	$27.42$	$85.59$	$25.548$	$26.815$	${\color[rgb]{0,0.88,0}\mathbf{1.137}}$	${\color[rgb]{0,0.88,0}\mathbf{0.182}}$	$0.240$
		15	$30.95$	$4.90$	$28.71$	$86.48$	$25.735$	$27.173$	$1.000$	$0.162$	$0.280$
		20	$32.81$	$5.07$	$29.40$	$86.34$	$25.807$	${\color[rgb]{0,0.88,0}\mathbf{27.249}}$	$0.947$	$0.154$	$0.309$
		25	$34.78$	$5.22$	$30.31$	$85.52$	${\color[rgb]{0,0.88,0}\mathbf{25.910}}$	$27.209$	$0.835$	$0.136$	$0.332$
	APG	10	$29.70$	${\color[rgb]{0,0.88,0}\mathbf{4.57}}$	${\color[rgb]{0,0.88,0}\mathbf{26.92}}$	$81.76$	$25.256$	$25.528$	${\color[rgb]{0,1,0}1.107}$	${\color[rgb]{0,1,0}0.178}$	${\color[rgb]{0,0.88,0}\mathbf{0.160}}$
		15	${\color[rgb]{0,0.88,0}\mathbf{29.46}}$	$4.70$	$27.36$	$83.78$	$25.612$	$26.136$	$1.058$	$0.172$	${\color[rgb]{0,1,0}0.168}$

Appendix

Table 7: Image quality for SDXL ( Podell et al., 2024 ) samples generated for ImageNet ( Russakovsky et al., 2015 ) and COCO ( Lin et al., 2014 ) datasets using CFG ( Ho & Salimans, 2022 ) , CACFG, and APG ( Sadat et al., 2025 ) . Best result for each dataset are given in bold green and second best in light green .

CACFG: Curvature-Aware Classifier-Free Guidance and Optimal Control

Abstract

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