2nd/10_Wiki/Topics/AI_and_ML/State-Space.md

id, title, category, status, canonical_id, aliases, duplicate_of, source_trust_level, confidence_score, verification_status, tags, raw_sources, last_reinforced, github_commit, tech_stack

id	title	category	status	canonical_id	aliases	duplicate_of	source_trust_level	confidence_score	verification_status	tags	raw_sources	last_reinforced	github_commit	tech_stack
wiki-2026-0508-state-space	State Space	10_Wiki/Topics	verified	self	ssm state-space-model mamba s4 linear-rnn	none	A	0.9	applied	state-space control-theory ssm mamba sequence-modeling		2026-05-10	pending	language framework python pytorch-mamba-jax

State Space

매 한 줄

"매 hidden state evolves over time, observations emerge from it". State space 는 매 control theory / signal processing 의 핵심 representation: x_{t+1}=Ax_t+Bu_t,\; y_t=Cx_t+Du_t. 매 2024-2026 의 deep learning 의 SSM (S4, Mamba, Mamba-2) 가 매 Transformer 의 long-context alternative 로 부상.

매 핵심

매 control-theory origin

• State x_t: 매 system 의 internal memory.
• Input u_t: 매 external control / observation.
• Output y_t: 매 measurable.
• Matrices A, B, C, D: 매 dynamics, input mapping, observation, feedthrough.
• Discrete vs continuous: \dot x = Ax + Bu (continuous) vs x_{t+1} (discrete).

매 deep learning SSM 진화

• HiPPO (2020): 매 long-range memory 의 polynomial 근사 — 매 A matrix 의 영리한 init.
• S4 (2022): structured SSM, FFT-based convolution view, long-range arena SOTA.
• H3, Hyena: 매 SSM + gating 의 hybrid.
• Mamba (2023): 매 selective SSM — 매 input-dependent A,B,C, hardware-aware parallel scan.
• Mamba-2 (2024): 매 SSD (state-space duality) — 매 attention 과 의 unification.
• Hybrid (2025+): 매 Jamba, Zamba, Samba — 매 Transformer + Mamba mix.

매 vs Transformer

측면	Transformer	SSM (Mamba)
Train	O(N²) parallel	O(N) parallel scan
Infer	O(N) per token, KV cache	O(1) per token, fixed state
Memory at infer	grows with context	constant
Long context	quadratic cost	linear cost
In-context recall	strong	weaker (improving with hybrids)

매 응용

1. Control systems — 매 robotics, aerospace, Kalman filter.
2. Time series — 매 Kalman / particle filter, dynamic factor model.
3. Long-context LLM — 매 Mamba-3B, Jamba-1.5 의 1M+ context.
4. DNA / genomics — 매 Caduceus, HyenaDNA 의 long sequence.
5. Audio — 매 SaShiMi 의 raw-waveform generation.

💻 패턴

Classic discrete state space (control)

import numpy as np
def simulate(A, B, C, D, u, x0):
    x, ys = x0.copy(), []
    for ut in u:
        y = C @ x + D @ ut
        x = A @ x + B @ ut
        ys.append(y)
    return np.array(ys)

Kalman filter

def kalman_step(x, P, u, z, A, B, C, Q, R):
    # predict
    x = A @ x + B @ u
    P = A @ P @ A.T + Q
    # update
    K = P @ C.T @ np.linalg.inv(C @ P @ C.T + R)
    x = x + K @ (z - C @ x)
    P = (np.eye(len(x)) - K @ C) @ P
    return x, P

S4 convolutional view (PyTorch)

import torch, torch.nn.functional as F
def s4_conv(u, A_bar, B_bar, C, L):
    # K_l = C A_bar^l B_bar  →  conv kernel of length L
    K = torch.stack([C @ torch.matrix_power(A_bar, l) @ B_bar for l in range(L)])
    y = F.conv1d(u.unsqueeze(0), K.flip(0).unsqueeze(0).unsqueeze(0))
    return y.squeeze()

Mamba selective scan (mamba-ssm package)

from mamba_ssm import Mamba
import torch
model = Mamba(d_model=512, d_state=16, d_conv=4, expand=2).cuda()
x = torch.randn(2, 1024, 512, device="cuda")
y = model(x)  # [B, L, D]  — O(L) parallel scan

HiPPO initialization (LegS)

def hippo_legs(N):
    A = np.zeros((N, N))
    for n in range(N):
        for k in range(N):
            if n > k: A[n, k] = -np.sqrt((2*n+1) * (2*k+1))
            elif n == k: A[n, k] = -(n + 1)
    return A

Mamba-2 SSD block (simplified)

def ssd_block(X, A, B, C, chunk=64):
    # state-space duality: structured matrix multiply
    # parallelizable as matmul, shows attention-SSM equivalence
    L = X.shape[1]
    Y = torch.zeros_like(X)
    state = torch.zeros(...)
    for s in range(0, L, chunk):
        Y[:, s:s+chunk], state = ssd_chunk(X[:, s:s+chunk], A, B, C, state)
    return Y

Hybrid (Jamba-style: Mamba + attention)

class JambaBlock(nn.Module):
    def __init__(self, d, use_attn=False):
        super().__init__()
        self.layer = MultiheadAttention(d, 8) if use_attn else Mamba(d_model=d)
        self.mlp = MoE(d) if use_moe else MLP(d)
    def forward(self, x): return self.mlp(self.layer(x) + x) + x

매 결정 기준

상황	Approach
Control system 의 design	classical SSM + LQR / Kalman
Time-series filtering	Kalman / particle filter
Long-context language (>128k)	Mamba / Jamba hybrid
In-context recall heavy	Transformer or hybrid (not pure Mamba)
Genomics 1M token	HyenaDNA / Caduceus
Audio raw waveform	SaShiMi / Mamba-Audio

기본값: 매 control 의 classical SSM. 매 long-context LM 의 Mamba-2 또는 Jamba 의 hybrid (pure Mamba 의 recall 약점 의 보완).

🔗 Graph

• 부모: Control-Theory · Sequence-to-Sequence-Models
• 변형: S4 · Mamba · Linear-RNN
• 응용: Kalman-Filter-and-State-Tracking · Time-Series-Analysis
• Adjacent: Transformer · 데이터 사이언스 및 ML 엔지니어링

🤖 LLM 활용

언제: SSM literature 정리, mamba-ssm boilerplate, control-system identification 의 sympy / numpy code. 언제 X: real-time safety-critical control (aerospace, medical) — 매 verified controller / formal methods 의 영역.

❌ 안티패턴

• HiPPO init 무시: random init Mamba 의 long-range 성능 추락. 매 proper A init 필수.
• Pure Mamba 의 in-context retrieval 기대: needle-in-haystack 약함. Hybrid 권장.
• No selective mechanism: 매 vanilla S4 의 modern Mamba 보다 약함 — input-dependent param 필수.
• Classical SSM 의 nonlinearity 무시: 매 real system 의 nonlinear — EKF / UKF / particle filter 사용.
• CUDA scan 미활용: 매 naive Python loop 의 100배 slow. mamba-ssm 의 official kernel 사용.

🧪 검증 / 중복

• Verified (Kalman 1960, Gu et al. S4 2022, Gu & Dao Mamba 2023, Dao & Gu Mamba-2 2024, Jamba 2024).
• 신뢰도 A.

🕓 Changelog

날짜	변경
2026-05-08	Phase 1
2026-05-10	Manual cleanup — control SSM + modern Mamba/Mamba-2/Jamba