Metric catalog

Every field the compute layer can produce, projected directly from the kernel registry: the producing kernel, its display formula, apply level, required input fields, and configurable parameters. See the per-category pages for derivations, conventions, and interpretation.

Matrix properties

Field(s)	Kernel	Formula	Level	Requires	Config
greater_dim	greater_dim	\(N = \max(m, n)\)	PARAMETER	weights	—
lower_dim	lower_dim	\(M = \min(m, n)\)	PARAMETER	weights	—
aspect_ratio	aspect_ratio	\(Q = N / M \ge 1\)	PARAMETER	greater_dim, lower_dim	—
weights_std	weights_std	\(\operatorname{std}(W)\)	PARAMETER	weights	—
weights_rand	weights_rand	\(W_{\mathrm{rand}} = \operatorname{reshape}(P\,\operatorname{vec} W)\)	PARAMETER	weights	seed=42

Spectral decomposition

Field(s)	Kernel	Formula	Level	Requires	Config
weights_lsvs, weights_svals, weights_rsvs	weights_svd	\(W = U\Sigma V^\top\)	PARAMETER	weights	allow_cuda=True
weights_rand_lsvs, weights_rand_svals, weights_rand_rsvs	weights_rand_svd	\(W_{\mathrm{rand}} = U\Sigma V^\top\)	PARAMETER	weights_rand	allow_cuda=True
esd	esd	\(\lambda_i = \sigma_i^2 / N\)	PARAMETER	weights_svals, greater_dim	—
esd_rand	esd_rand	\(\lambda_i^{\mathrm{rand}} = (\sigma_i^{\mathrm{rand}})^2 / N\)	PARAMETER	weights_rand_svals, greater_dim	—
max_weights_sval	max_weights_sval	\(\sigma_{\max} = \max_i \sigma_i\)	PARAMETER	weights_svals	—
max_weights_rand_sval	max_weights_rand_sval	\(\sigma_{\max}^{\mathrm{rand}} = \max_i \sigma_i^{\mathrm{rand}}\)	PARAMETER	weights_rand_svals	—
min_weights_sval	min_weights_sval	\(\sigma_{\min} = \min_i \sigma_i\)	PARAMETER	weights_svals	—
min_weights_rand_sval	min_weights_rand_sval	\(\sigma_{\min}^{\mathrm{rand}} = \min_i \sigma_i^{\mathrm{rand}}\)	PARAMETER	weights_rand_svals	—
esd_max	esd_max	\(\lambda_{\max} = \max_i \lambda_i\)	PARAMETER	esd	—
esd_rand_max	esd_rand_max	\(\lambda_{\max}^{\mathrm{rand}} = \max_i \lambda_i^{\mathrm{rand}}\)	PARAMETER	esd_rand	—
esd_min	esd_min	\(\lambda_{\min} = \min_i \lambda_i\)	PARAMETER	esd	—
esd_rand_min	esd_rand_min	\(\lambda_{\min}^{\mathrm{rand}} = \min_i \lambda_i^{\mathrm{rand}}\)	PARAMETER	esd_rand	—

Norms

Field(s)	Kernel	Formula	Level	Requires	Config
pl_alpha_norm	pl_alpha_norm	\(\sum_i \lambda_i^{\alpha_{\mathrm{PL}}}\)	PARAMETER	esd, pl_alpha	—
tpl_alpha_norm	tpl_alpha_norm	\(\sum_i \lambda_i^{\alpha_{\mathrm{TPL}}}\)	PARAMETER	esd, tpl_alpha	—
model_pl_alpha_norm	model_pl_alpha_norm	\(\langle \log_{10}\sum_i \lambda_i^{\alpha_{\mathrm{PL}}}\rangle\)	IN_MODEL	pl_alpha_norm	—
model_tpl_alpha_norm	model_tpl_alpha_norm	\(\langle \log_{10}\sum_i \lambda_i^{\alpha_{\mathrm{TPL}}}\rangle\)	IN_MODEL	tpl_alpha_norm	—
frob_norm	frob_norm	\(\lVert W\rVert_F = \sqrt{\sum_i \sigma_i^2}\)	PARAMETER	weights_svals	—
nuclear_norm	nuclear_norm	\(\lVert W\rVert_* = \sum_i \sigma_i\)	PARAMETER	weights_svals	—
l2_norm	l2_norm	\(\lVert W\rVert_2 = \sigma_{\max}\)	PARAMETER	max_weights_sval	—
log_norm	log_norm	\(\langle \log_{10}\lVert W\rVert_F^2\rangle\)	IN_MODEL	frob_norm	—
log_spectral_norm	log_spectral_norm	\(\langle \log_{10}\lVert W\rVert_2^2\rangle\)	IN_MODEL	l2_norm	—
param_norm	param_norm	\(\sum_\ell \lVert W_\ell\rVert_F^2\)	IN_MODEL	frob_norm	—
log_prod_frob_norm	log_prod_frob_norm	\(\sum_\ell \log_{10}\lVert W_\ell\rVert_F\)	IN_MODEL	frob_norm	—
log_prod_spectral_norm	log_prod_spectral_norm	\(\sum_\ell \log_{10}\lVert W_\ell\rVert_2\)	IN_MODEL	l2_norm	—

Ranks

Field(s)	Kernel	Formula	Level	Requires	Config
effective_rank	effective_rank	\(\exp\!\big(-\sum_i p_i \ln p_i\big),\ p_i = \sigma_i / \sum_j \sigma_j\)	PARAMETER	weights_svals	—
hard_rank	hard_rank	\(\#\{i : \lambda_i > \texttt{rtol}\cdot\lambda_{\max}\}\)	PARAMETER	esd	rtol=1e-05
mp_soft_rank	mp_soft_rank	\(\lambda_+ / \lambda_{\max}\)	PARAMETER	mp_esd_max, esd_max	—
stable_rank	stable_rank	\(\lVert W\rVert_F^2 / \lVert W\rVert_2^2\)	PARAMETER	frob_norm, l2_norm	—

Heavy-tailed fits

Field(s)	Kernel	Formula	Level	Requires	Config
expon_concentration	expon_concentration	\(\#\{i : \lambda_i \ge x_{\min}^{\mathrm{E}}\} / M\)	PARAMETER	esd, expon_esd_xmin	—
pl_concentration	pl_concentration	\(\#\{i : \lambda_i \ge x_{\min}^{\mathrm{PL}}\} / M\)	PARAMETER	esd, pl_esd_xmin	—
tpl_concentration	tpl_concentration	\(\#\{i : \lambda_i \ge x_{\min}^{\mathrm{TPL}}\} / M\)	PARAMETER	esd, tpl_esd_xmin	—
expon_presence	expon_presence	\((\lambda_{\max} - x_{\min}^{\mathrm{E}}) / (\lambda_{\max} - \lambda_{\min})\)	PARAMETER	esd_min, esd_max, expon_esd_xmin	—
pl_presence	pl_presence	\((\lambda_{\max} - x_{\min}^{\mathrm{PL}}) / (\lambda_{\max} - \lambda_{\min})\)	PARAMETER	esd_min, esd_max, pl_esd_xmin	—
tpl_presence	tpl_presence	\((\lambda_{\max} - x_{\min}^{\mathrm{TPL}}) / (\lambda_{\max} - \lambda_{\min})\)	PARAMETER	esd_min, esd_max, tpl_esd_xmin	—
expon_scale	expon_scale	\(\lambda_{\max}\,\Lambda_{\mathrm{E}}\)	PARAMETER	esd_max, expon_lambda	—
tpl_scale	tpl_scale	\(\lambda_{\max}\,\Lambda_{\mathrm{TPL}}\)	PARAMETER	esd_max, tpl_lambda	—
pl_alpha, pl_esd_xmin, pl_ks	power_law_fit	\(\hat{\alpha} = 1 + n_{\mathrm{tail}}\,\big/\sum_i \ln(\lambda_i / x_{\min})\)	PARAMETER	esd	fit_method=auto
pl_p_value [1]	pl_p_value	\(p = \Pr(D^* > D_{\mathrm{PL}})\)	PARAMETER	esd, pl_alpha, pl_esd_xmin, pl_ks	—
tpl_alpha, tpl_lambda, tpl_esd_xmin, tpl_ks	truncated_power_law_fit	\(p(\lambda) \propto \lambda^{-\hat{\alpha}}\,e^{-\hat{\Lambda}\lambda}\)	PARAMETER	esd	fit_method=auto
tpl_p_value [1]	tpl_p_value	\(p = \Pr(D^* > D_{\mathrm{TPL}})\)	PARAMETER	esd, tpl_alpha, tpl_lambda, tpl_esd_xmin, tpl_ks	—
expon_lambda, expon_esd_xmin, expon_ks	exponential_fit	\(\hat{\Lambda} = 1 / (\langle\lambda\rangle_{\ge x_{\min}} - x_{\min})\)	PARAMETER	esd	fit_method=auto
expon_p_value [1]	expon_p_value	\(p = \Pr(D^* > D_{\mathrm{E}})\)	PARAMETER	esd, expon_lambda, expon_esd_xmin, expon_ks	—

Marchenko-Pastur

Field(s)	Kernel	Formula	Level	Requires	Config
mp_esd_max, mp_esd_min, mp_bulk_std	marchenko_pastur_fit	\(\lambda_\pm = \sigma_{\mathrm{b}}^2\,(1 \pm 1/\sqrt{Q})^2\)	PARAMETER	esd_rand, esd_max, weights_std, aspect_ratio, lower_dim	—
mp_sval_max	mp_sval_max	\(\sigma_+^{\mathrm{MP}} = \sqrt{\lambda_+ N}\)	PARAMETER	mp_esd_max, greater_dim	—
mp_ks	mp_ks	\(D = \sup_\lambda \lvert \hat{F}(\lambda) - F_{\mathrm{MP}}(\lambda)\rvert\)	PARAMETER	aspect_ratio, mp_bulk_std, esd, mp_esd_max, mp_esd_min	—
mp_concentration	mp_concentration	\(\#\{i : \lambda_- \le \lambda_i \le \lambda_+\} / M\)	PARAMETER	esd, mp_esd_max, mp_esd_min	—
mp_presence	mp_presence	\((\lambda_+ - \lambda_-) / (\lambda_{\max} - \lambda_{\min})\)	PARAMETER	esd_min, esd_max, mp_esd_max, mp_esd_min	—
mp_num_spikes	mp_num_spikes	\(\#\{i : \lambda_i > \lambda_+\}\)	PARAMETER	esd, mp_esd_max	—

Tracy-Widom

Field(s)	Kernel	Formula	Level	Requires	Config
tw_esd_bound	tw_esd_bound	\(\lambda_{\mathrm{TW}} = \mu_{NM} + s_{NM}\,F_{\mathrm{TW}}^{-1}(1 - p)\)	PARAMETER	greater_dim, lower_dim, mp_bulk_std	p_value_threshold=0.005
tw_num_spikes	tw_num_spikes	\(\#\{i : \lambda_i > \lambda_{\mathrm{TW}}\}\)	PARAMETER	esd, tw_esd_bound	—

Alignment (cross-model)

Field(s)	Kernel	Formula	Level	Requires	Config
l_overlap	l_overlap	\(O^{L} = \lvert U_1^\top U_2\rvert\)	CROSS_MODEL	weights_lsvs, lower_dim	—
r_overlap	r_overlap	\(O^{R} = \lvert V_1^\top V_2\rvert\)	CROSS_MODEL	weights_rsvs, lower_dim	—
l_agreement	l_agreement	\((O^{L})_{ii}\)	CROSS_MODEL	l_overlap	—
r_agreement	r_agreement	\((O^{R})_{ii}\)	CROSS_MODEL	r_overlap	—
max_l_agreement	max_l_agreement	\(\max_j (O^{L})_{ij}\)	CROSS_MODEL	l_overlap	—
max_r_agreement	max_r_agreement	\(\max_j (O^{R})_{ij}\)	CROSS_MODEL	r_overlap	—
avg_l_agreement	avg_l_agreement	\(\big\langle (O^{L})_{ii}\big\rangle\)	CROSS_MODEL	l_agreement	—
avg_max_l_agreement	avg_max_l_agreement	\(\big\langle \max_j (O^{L})_{ij}\big\rangle\)	CROSS_MODEL	max_l_agreement	—
avg_max_r_agreement	avg_max_r_agreement	\(\big\langle \max_j (O^{R})_{ij}\big\rangle\)	CROSS_MODEL	max_r_agreement	—
avg_r_agreement	avg_r_agreement	\(\big\langle (O^{R})_{ii}\big\rangle\)	CROSS_MODEL	r_agreement	—

Model quality

Field(s)	Kernel	Formula	Level	Requires	Config
pl_alpha_weighted	pl_alpha_weighted	\(\alpha_{\mathrm{PL}}\,\log_{10}\lambda_{\max}\)	PARAMETER	esd_max, pl_alpha	—
w1_rand_distance	w1_rand_distance	\(\mathcal{W}_1(\lambda, \lambda^{\mathrm{rand}}) / \langle\lambda\rangle\)	PARAMETER	esd, esd_rand	—

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[1] (1,2,3)

Registered only when the taichi extra is installed; the bootstrap p-value uses the accelerated fitter.